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AT  RIVERSIDE 


THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


[UNIVERSITY    BULLETIN    No.    37.] 

ADDRESSES 

DELIVERED  BEFORE  Tin-: 

CALIFORNIA  TEACHERS'  ASSOCIATION, 

AT  RIVERSIDE, 

DECEMBER    28—31,    18Q1, 

BY 

Professors  in  the  University  of  California. 


i.  Educational  Progress  in  California. 

Prop.  Martin  Kellogg. 

2.  The  Social  Sciences  as  Aids  in  Teaching  History. 

Prof.  Bernard  Moses. 

3.  The  Past  and  Present  of  Elementary  Mathematics. 

Prof.  Irving  Stringham, 

4.  Physics  in  Secondary  Schools;  some  Aspects  of  the  Present 

Situation.  Prof.  Frederick  si  \u. 


BERK KLKY : 

PUBLISIIHD    P.Y    niK    UNIVERSITY. 
1892. 


Printed  at  the  State  Printing  Office,  Sacramento. 
A.  J.  Johxstox,  Superintendent. 


2,a 


EDUCATIONAL  PROGRESS  IN  CALIFORNIA. 

By  Professor  Martin  Kellogg, 

Acting  President  of  the  University  of  California. 


California  was  admitted  into  the  Union  in  1850.  Its 
preliminary  Constitutional  Convention  was  held  a  year 
earlier.  The  years  of  its  State  life  may  be  roughly 
divided  into  decades,  of  which  only  four  are  now  com- 
plete. This  division  of  time  will  afford  a  convenient 
method  of  grouping  the  events  of  our  educational 
history. 

I.  We  go  back,  first,  to  the  beginnings  of  our  school 
system.  The  foundation  for  this  system  was  laid  in 
1X49,  in  the  Constitutional  Convention  at  Monterey. 
An  inviolable  school  fund  was  then  established.  Article 
IX  of  the  Constitution  said:  "The  Legislature  shall 
encourage,  by  all  suitable  means,  the  promotion  of 
intellectual,  scientific,  moral,  and  agricultural  improve- 
ment." Section  4  of  the  same  article  directs  the  Legis- 
lature to  care  for  a  permanent  fund  for  the  support  of 
a  university  "  for  the  promotion  of  literature,  the  arts 
and  sciences." 

But  there  were  schools  antedating  1849.  As  early 
as  October,  1847,  the  Town  Council  of  San  Francisco 
provided  for  the  erection  of  a  small  school-house,  and 
in  April,  1848,  it  was  occupied,  with  Thomas  Douglass 
(a  Yale  graduate)  as  teacher  (salary  $1,000  a  year). 
There  were  at  first  six  pupils;  the  number  increased  to 
thirty-seven,  then  suddenly   dropped   to   eight.     The 

478245 


4  Riverside  Addresses. 

dropping  off  was  due  to  the  mining  excitement  which 
swept  so  many  off  to  the  interior,  and  among  them 
teacher  Douglass. 

During  these  years,  private  schools  were  opened  by 
Messrs.  Marston,  Williams,  and  Pelton. 

The  first  school  law  was  passed  by  the  Legislature  of 
1850-51.  Any  district  drawing  money  from  the  school 
fund  must  maintain  a  school  at  least  three  months  in 
the  y ear.  This  law  provided  for  the  establishment  of 
high  schools.  It  allowed  religious  schools  to  receive 
aid  from  the  fund. 

San  Francisco  had  had,  in  1850,  a  free  public  school, 
independent  of  any  State  law.  The  Common  Council 
established  it,  and  John  C.  Pelton  was  at  its  head. 

According  to  the  report  of  the  first  State  Superin- 
tendent, in  1 85 1,  there  were  about  six  thousand  chil- 
dren in  California  between  the  ages  of  4  and  18. 

II.  Det  us  now  pass  on  to  the  neighborhood  of  the 
sixties.  In  1858  there  were  over  forty  thousand  chil- 
dren of  school  age,  of  whom  less  than  one  half  attended 
the  public  schools.  In  1855,  religious  schools  had  been 
shut  out  from  the  school  fund.  In  1861,  an  attempt 
to  admit  them  again  to  its  benefits  was  defeated.  Much 
interest,  during  this  first  decade,  had  been  shown  in 
developing  the  school  system.  The  patrons  of  the 
schools  still  paid  rate-bills.  The  schools  were  not  yet 
in  a  condition  at  all  satisfactory  to  the  State  Superin- 
tendents, as  is  evinced  by  their  reports. 

I  turn  now  to  another  line  of  development,  viz.:  the 
beginnings  of  college  instruction.  If  I  give  this  a  large 
place,  I  plead  in  excuse  the  "  personal  equation,"  as  my 
environment  has  given  me  a  special  interest  in  college 
work. 


Educational  Progress  in   California.  5 

In  i860,  the  College  of  California  began  college 
instruction  in  Oakland.  It  was  not  the  first  college  on 
this  coast.  Other  colleges,  or  "universities,"  were 
earlier  in  the  field.  But  those  other  institutions 
depended  chiefly  on  what  we  call  preparatory  classes. 
Whoever  aspired  to  a  college  degree  must  take  his 
advanced  studies  as  he  could  get  them,  in  company 
with  scholars  of  lower  grade.  The  one  noteworthy- 
distinction  of  the  College  of  California  was  this:  It  had 
no  preparatory  school  work  to  look  after;  it  had  a 
Faculty  exclusively  devoted  to  college  studies.  Its 
standard  was  the  standard  of  the  Eastern  States.  This 
made  a  marked  advance  upon  any  previous  effort. 

From  the  first  years  of  California's  statehood,  a  few 
educated  men  had  ooked  forward  to  such  a  college. 
In  1849,  a  charter  was  sought  through  the  first  Legis- 
lature. The  bill  authorizing  such  charters  was  passed, 
leaving  the  special  application  to  be  made  to  the 
Supreme  Court.  A  financial  basis  of  $20,000  was  to 
be  a  prerequisite.  A  cluster  of  men,  afterward  inter- 
ested in  the  College  of  California,  had  obtained  a 
promise  of  certain  tracts  of  land  and  applied  for  a 
charter,  but  defective  titles  defeated  the  application. 
This  was  in  1850. 

In  1853,  Henry  Durant  came  to  do  educational  work 
in  California,  and  started  a  preparatory7  school  in  Oak- 
land. He  interested  a  few  college  men  in  his  enter- 
prise, which,  from  the  outset,  aimed  at  the  building  up 
of  a  college.  A  small  building  was  rented  at  a  high 
price,  and  his  school  began  with  three  pupils.  In  1855 
the  three  had  increased  to  sixty. 

The  same  year  a  charter  for  a  college  was  obtained, 
the  Trustees  named  being  Fred.  Billings,  Sherman  Day, 
S.  H.  Willey,  T.  D.  Hunt,  M.  Brumagim,  E.  B.  Wals- 


6  Riverside  Addresses. 

worth,  J.  A.  Benton,  E.  McLean,  H.  Durant,  F.  W. 
Page,  R.  Simson,  A.  H.  Wilder,  and  S.  B.  Bell. 

These  Trustees  had  very  definite  aims.  One  of  these 
was  to  unite  the  religious  people  of  the  State  in  an 
effort  to  establish  a  first-class  college,  which,  like  Yale 
and  Princeton,  should  be  Christian  in  tone,  but  practi- 
cally unsectarian.  Starr  King  was  a  member  of  the 
Board,  and  after  him  Horatio  Stebbins.  The  very 
breadth  of  their  scheme  increased  the  difficulties  of  the 
Trustees.  Devoted  denominationalists  could  not  be 
deeply  interested,  and  waited  for  an  institution  all 
their  own. 

Another  definite  aim  was  to  place  the  college  on  a 
high  footing  by  obtaining  for  a  President  some  distin- 
guished man  from  the  East.  Dr.  Horace  Bushnell  was 
almost  secured.  He  was  one  of  the  loftiest  souls  in  the 
whole  land,  and  was  deeply  interested  in  education. 
He  came  to  this  State  for  his  health,  and  spent  much 
time  in  looking  up  a  good  college  site.  With  health 
regained,  he  yielded  to  the  importunities  of  his  old 
church,  and  was  lost  to  California.  Drs.  Shedd  and 
R.  D.  Hitchcock  in  turn  declined  the  offers  of  the 
Trustees. 

Thrown  back  upon  their  own  efforts,  the  Board 
moved  forward.  They  secured  a  college  site,  now 
occupied  by  the  University  at  Berkeley,  and  dedicated 
-it  on  April  16,  i860.  In  the  summer  of  that  year  the 
college  began  its  work  in  Oakland.  Its  first  Freshman 
Class  had  been  trained  by  Dr.  Durant.  The  first  pro- 
fessors chosen  were  Durant  and  Kellogg,  the  latter  of 
whom  began  work  in  1 86 1 .  Prof.  Brayton  took  certain 
classes,  but  gave  most  of  his  energies— too  unspar- 
ingly—to the  school  left  by  Dr.  Durant. 

Thus,  after  the  first  ten  or  twelve  years  of  California's 


Educational  Progress  in  California.  7 

life,  we  see  the  full  establishment  of  a  common  school 
system,  and  the  beginning  of  separate,  determined 
college  work. 

III.  We  pass  on  to  the  seventies,  to  the  close  of  a 
second  decade.  What  advances  have  these  years  to 
show? 

On  the  part  of  the  schools,  a  great  expansion  and  a 
higher  level.  Early  in  the  sixties,  increased  provision 
was  made  for  a  school  fund.  John  Swett  became  State 
Superintendent,  and  argued  strongly  for  a  special  State 
tax.  Such  a  tax  was  laid  in  1864.  The  year  ending 
with  June,  1867,  "marks  the  transition  period  of  Cali- 
fornia from  rate-bill  common  schools  to  an  American 
free  school  system."  The  Superintendent  records  this 
fact  with  pardonable  pride.  A  school  library  system 
was  provided  by  the  law  of  1866,  and  put  into  success- 
ful operation.  The  average  salary  of  male  teachers 
was  reported  as  $77  a  month;  of  female  teachers,  $64. 
Fifty  thousand  children  were  enrolled  in  the  schools. 
From  1 86 1  to  1871  there  were  eight  State  Institutes. 
At  the  third  of  these,  called  in  San  Francisco  by  Super- 
intendent Swett,  there  was  an  attendance  of  four  hun- 
dred and  sixty-three.  In  1867,  five  hundred  teachers 
were  present.  These  Institutes  were  of  much  value  in 
giving  the  teachers  an  esprit  de  corps,  and  in  influenc- 
ing school  legislation.  In  1869,  the  State  tax  was 
increased  to  10  cents  on  each  $100. 

On  the  side  of  the  higher  education  there  was  an 
advance  from  the  college  to  the  university.  As  in 
the  case  of  the  college,  the  university  idea  had  long 
been  cherished:  how  it  should  take  shape,  was  a  ques- 
tion to  be  determined  by  events.  As  has  been  men- 
tioned, the  Constitution  of  1849  made  reference  to  a 


8  Riverside  Addresses. 

university,  and  enjoined  on  the  Legislature  the  protec- 
tion of  funds  granted  for  such  an  institution.  In  1856, 
Superintendent  Hubbs  urged  the  establishment  of  a 
university.  His  successor,  Superintendent  Moulder, 
discussed  the  subject  at  large. 

In  1853,  Congress  granted  to  this  State  seventy-twc 
sections  of  land  "  for  the  use  of  a  seminary  of  learning.' 
Also,  ten  sections  of  land  for  a  public  building  fund. 
In  1862,  by  the  Act  known  as  the  Morrill  Act,  Con- 
gress gave  to  the  State  one  hundred  and  fifty  thousand 
acres  of  public  land  for  an  Agricultural  and  Mechanic 
Arts  College.  This  donation  was  accepted  in  1864.  In 
the  previous  year  the  Legislature  appointed  a  Commis- 
sion to  report  a  plan  for  a  university,  the  chairman 
being  Prof.  J.  D.  Whitney.  The  scheme  elaborated  by 
this  Commission  was  very  different  from  the  plan  finally 
adopted.  In  1866  a  Board  of  Directors  was  established 
for  an  "Agricultural,  Mining,  and  Mechanic  Arts  Col- 
lege." The  next  year  the  Directors  located  the  college 
in  Alameda  County. 

In  the  same  year,  1867,  came  a  proposition  from  the 
College  of  California,  looking  to  a  larger  and  broader 
institution,  a  real  university.  The  Trustees  of  the 
college  had  tried  faithfully  to  expand  their  own  insti- 
tution; but  the  process  was  slow,  and  the  financial 
prospects  unfavorable  for  the  early  fulfillment  of  their 
hopes.  The}'  had  a  fine  property  in  Oakland,  and  the 
magnificent  site  of  one  hundred  and  sixty  acres  at 
Berkeley.  They  had  established  a  high  college  stand- 
ard. Through  their  seven  j-ears  of  actual  college 
work  at  Oakland,  and  through  the  very  successful 
Alumni  gatherings  of  men  from  Eastern  institutions,  a 
wider  interest  in  college  instruction  had  been  developed. 
The  Trustees  now  offered  to  turn  over  their  property 


Educational  Progress  in   California.  9 

and  good-will  to  a  State  University,  with  the  condition 
that  the  University  should  maintain  certain  colleges  of 
science  "  and  an  academical  college,"  "  all  of  the  same 
grade  and  with  courses  of  instruction  equal  to  those  of 
Eastern  colleges."  This  offer  was  accepted  by  the 
Board  of  Directors  of  the  Agricultural  College.  An 
act  to  create  and  organize  the  University  of  California 
was  introduced  in  the  lower  house  of  the  Legislature 
March  5,  1868,  by  John  W.  Dwindle,  and  approved 
March  23d.     We  call  that  our  Charter  Day. 

The  College  of  California  continued  its  instruction 
till  the  summer  of  1869,  and  then  became  merged  into 
the  University.  The  first  appointees  to  professorships 
in  the  University  were  Dr.  John  Le  Conte  and  Prof. 
Kellogg.  Dr.  Joseph  De  Conte  and  others  soon  fol- 
lowed. The  first  University  Class  entered  in  1869.  In 
1870  there  were  two  classes,  and  young  women  had 
already  been  admitted  on  an  equal  footing  with  young 
men. 

The  end  of  this  second  decade,  therefore,  is  marked 
by  the  complete  establishment  of  a  free  public  school 
system ;  and  by  the  establishment  of  a  State  Univer- 
sity, taking  up  and  much  enlarging  the  previous  work 
of  the  college.  New  attention  was  given  to  the  claims 
of  the  natural  sciences,  to  the  demand  for  training  for  the 
scientific  professions,  and  to  the  bearing  of  education 
on  industrial  occupations.  Military  training,  the  only 
condition  imposed  with  the  national  grant,  became  a 
part  of  the  students'  education.  Our  State  took  a  step 
in  advance  of  any  other  in  making  tuition  in  its  Uni- 
versity absolutely  free. 

IV.     The  third  decade  brings  us  to  the  eighties. 
In  these  years  the  public  school  system  gained  in 


10  Riverside  Addresses. 

efficiency.  In  his  report  for  1875,  State  Superintendent 
Bolander  notes  an  important  advance  in  furnishing  aid 
to  the  smaller  school  districts.  Some  of  them  had 
fallen  short  of  the  minimum  of  three  months  :  now 
the  distribution  was  more  equal  and  helpful.  He 
argues  in  this  report  for  a  special  and  extended  course 
in  the  University  for  teachers. 

During  this  decade  there  was  a  local  political  cyclone, 
which  for  awhile  overturned  the  calculations  of  the 
old  parties.  It  brought  about,  in  1879,  a  new  Consti- 
tutional Convention,  whose  proceedings  interest  us  in 
one  or  two  points.  First,  high  schools  were  left  out  of 
the  schools  to  be  supported  by  the  State  tax.  Any 
community  wishing  for  a  high  school  must  itself  meet 
the  expense.  This  omission  of  the  high  school,  in 
the  series  of  schools  provided  by  the  State,  has  had 
a  chilling  effect  on  the  higher  education.  Common 
schools  were  free,  the  University  was  free  ;  but  how 
were  pupils  to  get  from  the  one  to  the  other?  The 
larger  cities  bridged  the  chasm  without  much  hesita- 
tion by  establishing  high  schools  at  their  own  expense. 
The  smaller  towns  and  country  districts  were  slow  to 
assume  the  burden. 

Notwithstanding  these  difficulties,  the  University 
maintained  its  high  standard  of  admission.  It  could 
not  lower  the  requirements  without  taking  a  long  and 
fatal  step  backward. 

That  Constitutional  Convention  took  another  step 
which  was  not  adverse,  but  favorable  to  the  State  Uni- 
versity. Article  IX,  Section  9,  declared:  "The  Uni- 
versity of  California  shall  constitute  a  public  trust,  and 
its  organization  and  government  shall  be  perpetually 
continued  in  the  form  and  character  prescribed  by  the 
organic  act  creating  the  same."     In  that  radical  Con- 


Educational  Progress  in   California.  11 

vention  it  was  decided  that  the  University  was  some- 
thing not  to  be  uprooted. 

V.     And  now  we  come  to  the  nineties. 

The  common  schools  have  prospered  as  never  before. 
More  really  well-educated  teachers  have  been  found  in 
the  profession.  As  a  profession,  teaching  has  become 
a  more  honorable  and  a  more  permanent  occupation. 
We  have  now  three  normal  schools  instead  of  one. 
Teachers'  Institutes  have  been  made  a  part  of  the 
school  system,  and  they  have  been  attended  with 
increasing  interest,  even  enthusiasm.  The  whole  ma- 
chinery of  the  school  system  has  been  elaborated  and 
made  more  efficient.  The  teachers  have  shown  increas- 
ing devotion  to  their  work,  and  the  best  of  them  have 
found  appreciation  in  the  community  at  large.  The 
one  weak  point  in  the  system,  showing  itself  chiefly 
in  the  larger  cities,  is  the  method  of  choosing  Boards 
of  Education. 

The  unfortunate  chasm  between  the  common  school 
and  the  University  has  been  more  and  more  regretted, 
and  there  have  been  strenuous  efforts  to  bridge  it  over. 
Years  ago  the  schools  were  authorized  to  add  some- 
what to  their  courses,  in  the  direction  of  the  scientific 
requirements  of  the  University.  The  last  Legislature 
passed  two  bills  for  the  establishment  of  high  schools: 
the  one  for  county  high  schools,  the  other  for  city  or 
town  high  schools  and  union  district  high  schools. 
The  burden  of  maintenance  was  still  to  be  local,  but 
provision  was  made  for  putting  the  question  before  the 
people,  and  for  levying  the  needful  tax.  These  laws 
have  spurred  up  a  considerable  number  of  communities, 
and  the  work  of  establishing  high  schools  is  now  going 
forward  with  the  most  favorable  prospects. 


12  Riverside  Addresses. 

The  standard  aimed  at  in  these  high  schools  is  most 
encouraging.  As  one  object  in  establishing  them  was 
to  pave  the  way  to  the  University,  the  requirements  of 
the  University  have  been  cheerfully  met.  These  re- 
quirements do  not  hinder,  but  rather  help,  in  fixing  a 
curriculum  of  most  value  to  students  who  take  no 
further  course.  It  is  always  borne  in  mind  that  only 
a  minority  are  to  take  a  college  course,  but  this  minor- 
ity must  not  be  debarred  from  going  forward. 

A  special  feature  of  the  last  decade  has  been  the 
attainment  of  a  uniform  standard.  This  has  been 
brought  about  by  a  system  of  accrediting.  Our  Uni- 
versity was  not  the  first  to  devise  this  system:  Michigan 
was  already  using  it.  But  we  may  claim  to  have  made 
the  system  more  thorough  than  we  found  it.  More 
pains  is  taken  here  than  anywhere  else  to  visit,  advise, 
and  encourage  each  one  of  the  main  departments  of 
instruction  in  every  school  applying  for  recognition. 
The  results  have  been  very  satisfactory,  and  the  circle 
of  accredited  schools  is  constantly  widening.  Nearly 
thirty  are  already  on  the  list. 

Another  gain  is  seen  in  the  call  for  a  longer  curricu- 
lum in  the  high  schools.  Californians  have  been  fond 
of  short  cuts  and  the  briefest  possible  courses  of  study. 
But  the  State  is  less  in  a  hurry  than  it  was.  It  is 
coming  to  see  that  haste  makes  waste,  that  the  basis  of 
a  broad  and  solid  scholarship  cannot  be  laid  by  a  fever- 
ish urgency.  A  few  high  schools  have  already  provided 
a  four  years'  course.  If  this  can  become  general,  it  will 
react  on  the  grammar  schools,  and  bring  about  in  them 
a  reduction  of  work  and  an  earlier  advance  to  the  high 
school  point.  Thus  those  who  are  on  their  way  to  the 
college  will  lose  no  time;  but  they  will  have  changed 
some  of  the  needless  repetitions  or  premature  exactions 


Educational  Progress  in   California.  13 

of  the  grammar  years  for  the  more  careful  and  thorough 
studies  of  the  high  school. 

The  State  University  has  made  good  progress  during 
this  last  decade,  notwithstanding  the  difficulties  as  to 
preparation.  Some  academies  and  private  schools  have 
helped  to  fill  the  place  of  the  missing  high  schools. 
Liberal  aid  has  been  given  by  the  State  to  its  University 
by  a  continuing  tax.  The  staff  of  instruction  has  been 
much  enlarged,  with  a  corresponding  expansion  of  the 
courses  offered  to  the  students.  Special  aid  has  come 
to  the  Agricultural  and  Mechanic  Arts  courses  from 
a  supplementary  Morrill  Aid  Fund.  The  professional 
schools  connected  with  the  University  have  a  much 
larger  number  on  their  rolls.  Denominational  colleges 
and  schools  have  sprung  up  at  various  points  in  the 
State. 

So  much  for  the  history  of  this  fourth  decade.  The 
fifth  is 'but  beginning.     But 

VI.  The  new  decade  opens  with  extraordinary 
promise  of  educational  development.  Three  things  are 
especially  worthy  of  note: 

i.  The  springing  into  life  of  a  new  and  powerful 
university.  I  need  not  say  where  it  is  situated,  nor 
what  name  it  bears.  The  people  of  California  have  all 
heard  of  it,  and  have  all  been  interested  in  its  brilliant 
beginnings.  To  our  educational  force  in  this  State  an 
entire  new  Faculty  has  been  added  by  the  magic  wand 
of  wealth.  This  corps  of  able  scholars  and  accom- 
plished teachers  has  become  known  already  in  all  our 
educational  circles.  You  welcome  some  of  their  num- 
ber at  this  meeting  of  the  State  Association. 

2.  A  second  important  event  of  the  3^ear  1891  is  the 
movement  for  the  University  Extension.     It  is  ahead}'- 


14  Riverside  Addresses. 

general,  and  promises  to  become  almost  overpowering. 
It  is  not  endemic,  but  epidemic,  affecting  the  whole 
country,  and  coming  almost  as  swiftly  as  the  grippe. 
Earlier  and  more  slowly  it  grew  up  in  Great  Britain. 

But  no  such  movement  is  capriciously  causeless.  In 
our  own  country  there  was  a  deliberate  and  widespread 
effort  to  popularize  knowledge,  through  the  Chautauqua 
movement.  That  movement  reached  a  multitude  of 
earnest  minds,  and  has  made  itself  felt  in  this  State. 
When  a  higher  plan  came  in,  under  the  name  of  Uni- 
versity Extension,  it  found  the  public  ready  and  eager 
to  welcome  it. 

Just  what  form  University  Extension  is  to  take  in 
California,  it  is  too  early  to  judge.  If  it  can  be  kept 
on  a  true  university  level,  it  will  do  much  to  carry  the 
higher  instruction  to  those  who  cannot  attend  college 
classes.  In  that  case,  it  will  call  for  a  large  increase 
of  the  University  Faculties.  The  field  work  will  rival 
in  its  demands  the  proper  home  work  of  the  univer- 
sities. If  too  many  places  are  clamorous  for  their  share 
of  attention,  there  will  be  encouragement  for  some 
adventurers,  coming  in  by  doubtful  doors,  who  will  not 
satisfy  the  more  thoughtful  minds.  There  are  many 
problems  still  unsettled  in  the  application  of  this  sys- 
tem to  our  communities.  But  some  form  or  forms  of 
UniversityExtension  will  surely  play  a  prominent  part 
in  California  in  this  last  decade  of  the  nineteenth 
century. 

3.  There  is  much  more  demand  for  distinctive  courses 
of  instruction  in  the  interest  of  the  teaching  profession. 
Normal  schools  there  have  been  for  many  years.  Now 
there  is  a  call  for  a  similar  and  higher  professional 
training  in  the  Universities.  Chairs  of  pedagogy  have 
been  established  in  nianv  institutions.     Harvard  has  a 


Educational  Progress  in  California.  15 

new  Professor  of  the  History  and  the  Art  of  Teaching. 
Yale  gives  several  courses  in  pedagogy.  The  univer- 
sities of  this  State  recognize  this  as  an  important 
department  of  instruction. 


This  review  of  educational  progress  in  California  is 
necessarily  very  incomplete;  but  it  shows  that  no  cata- 
clysm has  taken  place  in  our  educational  history. 
During  all  these  decades  there  have  been  forces  at 
work  for  the  promotion  of  the  educational  interests  of 
the  State.  If  these  interests  are  now  prosperous,  it  is 
in  sequence  of  previous  long-continued  and  earnest 
efforts.  If  there  is  a  harvest  for  the  teachers  of  Cali- 
fornia to  reap  in  the  near  future,  it  is  because  some 
patient  husbandmen  have  sown  the  seed. 

We  that  are  older  almost  envy  the  young  teachers 
who  are  to  take  part  in  this  new  and  larger  progress. 
But  there  is  a  certain  satisfaction  in  having  helped  to 
lay  foundations.  And  let  me  utter  a  note  of  warning. 
If  in  these  coming  years  the  teachers  lose  their  love 
for  their  work,  if  they  forget  the  self-sacrifice  which 
that  work  implies,  and  cease  to  emulate  the  single- 
hearted  devotion  of  the  pioneers,  if  they  seek  rather 
for  easy  and  comfortable  places,  then  true  educational 
progress  will  cease.  Young  ladies  and  gentlemen,  you, 
too,  must  be  pioneers  for  a  still  better  and  brighter 
future.  You,  too,  must  help  to  lay  foundations  which 
shall  be  out  of  sight  and  forgotten,  it  may  be,  that  the 
future  temple  of  education  may  rise,  sacred  and  glorious, 
in  this  home  of  our  love  and  our  hopes. 


16  Riverside  Addresses. 


THE  SOCIAL  SCIENCES  AS  AIDS  IN  TEACHING 
HISTORY. 

By  Bernard  Moses, 

Professor  of  History  arid  Political  Economy. 


During  the  few  moments  in  which  I  may  engage 
your  attention,  I  shall  invite  you  to  consider  the  social 
sciences  as  aids  in  teaching  history.  A  certain  justifi- 
cation for  bringing  this  matter  before  you  may  be  found 
in  the  position  which  history  and  kindred  subjects  now 
occupy  in  the  general  scheme  of  academic  studies. 
What  this  position  is  may  be  seen  by  glancing  at  the 
historical  development  of  the  curriculum. 

Considered  with  reference  to  its  growth,  the  primary 
phase  of  the  academic  curriculum  is  that  in  which 
religion  is  the  sole  topic  embraced.  The  Mohamme- 
dans, who  found  the  Alexandrian  library  superfluous, 
in  so  far  as  its  books  agreed  with  the  Koran,  and  use- 
less, in  so  far  as  they  differed  from  it,  were  the  products 
and  the  advocates  of  this  simple  curriculum.  It 
appeared  again  in  the  Middle  Ages,  when  the  Christian 
church,  in  its  pretensions  to  omniscience  and  absolute 
authority,  essayed  to  fix  the  limits  of  human  inquiry. 
Galileo  might  not  teach  that  the  earth  moves,  because 
this  proposition  was  not  in  the  predetermined  circle  of 
knowledge. 

The  first  considerable  extension  of  the  realm  of 
knowledge  was  made  through  that  great  intellectual 


Social  Sciences  in    Teaching  History.  17 

movement  which  we  call  the  revival  of  learning.  This 
opened  to  modern  minds  a  buried  and  almost  for- 
gotten civilization,  a  civilization  whose  art  and  poetry 
and  philosophy  transcended  the  highest  conceptions  of 
those  to  whom  it  was  revealed.  The  study  of  antiquity 
drew  to  itself  a  constantly  increasing  number  of 
scholars,  and,  in  their  enthusiasm  for  ancient  learning, 
all  other  subjects  remained  in  comparative  neglect. 
But  on  this  study  of  ancient  literature  and  ancient 
institutions,  the  church  looked  with  suspicion.  It 
feared  the  anti-christian  influence  of  pagan  thought. 
It  feared  a  revival,  not  only  of  the  learning,  but  also 
of  the  worship  of  paganism.  It  feared  the  loss  of 
stability  and  power  in  the  presence  of  the  unshackled 
mind,  and,  therefore,  the  means  for  developing  the  new 
learning  had  to  grow  up  outside  of  the  cloister  and 
find  other  supporters  than  the  clergy.  The  secular 
authorities  in  states  and  cities  then  became  the  patrons 
of  learning  and  the  founders  of  schools.  The  oldest 
universities  of  Italy  and  Germany  were  established 
under  the  impulse  of  this  spirit;  and  thus  the  demand 
for  instruction  in  the  new  field,  which  the  church  would 
not  satisfy,  was  met  by  the  wise  liberality  of  princes 
acting  in  behalf  of  the  states  whose  secular  power 
they  wielded.  With  the  development  of  the  new 
learning,  theological  teaching  came  out  of  the  cloisters 
and  was  established  at  the  universities.  The  church 
ceased  to  be  the  sole  patron  of  education,  and  scholasti- 
cism no  longer  covered  the  whole  field  of  thought  and 
inquiry.  A  new  topic  was  added  to  the  curriculum  of 
human  knowledge.  To  the  study  of  divinity  was  added 
the  study  of  the  humanities. 

Through  the  influence  of  the  attractive  revelations 
of  antiquity  and  the  powerful  prejudice  of  the  church, 
2 — A 


18  Riverside  Addresses. 

the  early  discoveries  in  the  realm  of  nature  either 
suffered  neglect  or  were  regarded  as  unfit  material  for 
academic  teaching.  But  in  the  first  half  of  this  century 
was  revealed  the  second  great  intellectual  revival  of 
modern  times.  The  fragmentary  and  unsystematic 
investigations  of  earlier  scientists  grew  into  a  system- 
atic and  well-ordered  search  for  scientific  truth,  and 
the  results  of  this  search,  not  less  wonderful  to  the 
nineteenth  century  than  were  the  revelations  of  the 
literary  revival  to  the  fifteenth,  have  grown  into  that 
body  of  knowledge  which  we  call  modern  science;  and 
this  science,  opposed  at  first  by  the  church  and  neg- 
lected by  the  humanists,  now  takes  its  place  by  the 
side  of  scholasticism  and  the  humanities  in  the  great 
universities  of  the  world,  and  forms  a  new  addition  to 
the  field  of  learning.  Thus  to  the  primitive  curriculum 
embracing  the  subjects  properly  involved  in  the  study 
of  religion,  each  of  the  ages  in  question  has  added  a 
new  topic  or  a  new  department  of  academic  instruction. 
One  has  added  the  humanities;  the  other,  modern 
science. 

In  these  last  decades,  a  fourth  topic,  or  department 
of  study,  has  been  added  to  the  list,  and  it  could  not 
well  have  been  added  earlier.  During  the  seventeenth 
century  and  the  early  part  of  the  eighteenth,  the 
period  of  European  absolutism,  when  society  was  sup- 
posed to  be  directed  arbitrarily  by  the  capricious  will 
of  an  absolute  ruler,  there  was  little  incentive  to  study 
the  history  of  society  or  to  attempt  to  discover  the  laws 
of  its  activity  and  growth.  But  by  degrees  it  came  to 
be  generally  acknowledged  that  the  ultimate  political 
power  in  the  state,  that  power  which  is  the  source  of 
all  delegated  authority,  rests  with  the  people.  So  long 
as  the   people  had  no  share   in  the  political  control 


Social  Sciences  in   Teaching  History.  19 

of  society,  their  knowledge  or  their  ignorance  of  the 
principles  and  practices  of  public  administration  was  a 
matter  of  little  concern.  There  was  no  inducement  for 
them  to  acquire  knowledge  of  the  details  of  govern- 
ments in  which  they  had  no  voice ;  but  when  the 
power  to  decide,  either  directly  or  indirectly,  all  ques- 
tions of  public  interest  devolved  upon  the  people,  it 
became  a  matter  of  grave  importance  that  they  should 
be  instructed  in  the  history  of  society  and  in  the  laws 
and  principles  which  underlie  its  organization,  activity, 
and  growth.  There  arose,  therefore,  a  demand  for  an 
enlargement  of  the  field  of  instruction,  which  should 
embrace  especially  the  subjects  .of  history,  economics, 
and  politics  ;  and  this  group  of  subjects  constitutes  the 
latest  important  addition  to  the  curriculum  of  academic 
studies. 

During  the  last  forty  years  this  department  of  knowl- 
edge has  become  more  and  more  important  in  the 
academic  instruction  in  America,  and  the  leading  uni- 
versities have  yielded  to  it  a  wider  and  still  increasing 
place  among  the  several  courses  of  study.  Many  of 
us,  still  not  very  old  men,  remember  distinctly  the  time 
when  history  was  not  considered  of  sufficient  impor- 
tance in  the  curriculum  to  demand  the  exclusive  atten- 
tion of  an  instructor.  It  was  turned  over  somewhat 
indiscriminately  to  any  member  of  the  Faculty  who 
might  happen  to  have  fewer  and  less  exacting  engage- 
ments than  the  others.  It  was  not  thought  necessary 
to  consider  his  attainments  ;  any  one  who  could  read 
and  understand  a  plain  narrative  ought  to  be  able  to 
teach  history.  What  changes  the  period  of  a  genera- 
tion has  wrought  in  the  academic  standing  of  these 
subjects  may  be  seen  by  observing  their  present  place 
in  the  curriculum  of  some  of  the  leading  universities 


20  Riverside  Addresses. 

of  this  country  :  Harvard  has  forty  courses  taught  by 
twelve  instructors  of  different  grades ;  Yale  thirty 
courses,  with  ten  instructors ;  Columbia  thirty-one 
courses,  with  eleven  instructors ;  Cornell  twenty- 
seven  courses,  with  ten  instructors ;  University  of 
Michigan  thirty  courses,  with  nine  instructors  ;  Uni- 
versity of  California  twenty-two  courses,  with  four 
instructors. 

An  examination  of  these  courses  will  reveal  the  fact 
that  while  the  several  studies  of  the  group  dealing  with 
historical,  economical,  and  political  phenomena  bear 
different  names,  they  are  all  nevertheless  designed  to 
contribute  to  a  common  end,  and  that  is  to  render  a 
rational  armnnt  of  society.  The  whole  group,  as  now 
constituted,  is  in  some  sense  an  outgrowth  of  the  course 
of  instruction  in  history  as  primarily  organized.  Its 
purpose  is  an  enlargement  of  the  purpose  of  instruction 
in  history.  As  formerly  presented,  history  showed  us 
somewhat  of  the  activity  of  a  nation,  but  jt  left  much 
to  be  known  as  to  the  nature  and  organization  of  the. 
nation.  To  supplement  the  knowledge  of  society 
which  history  furnished,  the  science  of  economics  was 
introduced,  which  undertook  to  give  a  scientific 
account  of  the  industrial  and  col  .nercial  organization 
and  activity  of  the  nation;  while  a  little  later  political 
science  came  to  enlarge  our  knowledge  of  man's  polit- 
ical relations  and  activities. 

All  the  subjects  arranged  in  the  curriculum  under 
these  three  general  heads  of  history,  economics,  and 
politics  will  be  found,  on  thorough  examination,  to  be 
closely  united  by  lines  of  mutual  dependence.  History, 
as  the  comprehensive  record  of  the  internal  affairs  and 
foreign  relations  of  nations,  necessarily  involves  discus- 
sions of  the  subject-matter  not  only  of  economics,  but 


Social  Sciences  in   Teaching  History.  21 

also  of  politics.  The  internal  affairs  considered  are 
not  simply  those  relating  to  distinguished  persons,  on 
whom  has  fallen  by  the  accident  of  birth  the  nominal 
guidance  of  the  public  administration ;  rather  that 
part  of  internal  or  domestic  history  which  relates  to 
the  industrial  and  commercial  order  and  progress  of  a 
nation.  The  popular  side  of  national  life  has  become 
especially  conspicuous  during  the  present  century,  for 
two  reasons: 

i .  Because  of  the  unprecedented  development  of  the 
commercial  and  industrial  interests  of  society. 

2.  Because  of  the  transfer  of  political  power  from 
an  irresponsible  head  of  the  state  to  the  body  of  the 
nation. 

These  changes  have  turned  attention  to  the  people, 
and  led  the  historian  to  emphasize  especially  popular 
interests  and  enterprises.  These  phases  of  national 
life  have  become  so  conspicuous  that  it  has  been  found 
to  be  desirable  to  rewrite,  from  a  new  point  of  view,  the 
history  of  certain  nations,  particularly  those  preeminent 
in  industry  and  commerce.  The  history  of  England, 
therefore,  in  its  latest  version,  under  the  hand  of  Mr. 
Green,  becomes  a  "history  of  the  English  people." 
Another  consequent  of  the  conspicuous  position  to 
which  the  people  have  risen  in  this  century  is  the 
development  of  studies  in  civilization,  whether  in  the 
form  of  the  German  Culturgeschichte,  or  the  French 
and  English  history  of  civilization,  or  studies  on  the 
nature  and  progress  of  society.  The  internal  history 
of  any  nation  now  demanded  from  the  teacher  is  that 
which  includes  not  only  the  phenomena  of  trade  and 
industry,  of  mone)^  and  taxation,  but  also  the  whole 
body  of  facts  from  the  life  of  the  people,  which  consti- 
tute the  data  of  economics,  the  phenomena  which  it  is 


22  Riverside  Addresses. 

the  business  of  economics  to  explain.  In  order,  there- 
fore, to  understand  that  part  of  a  nation's  history  which 
is  here  referred  to,  one  has  need  to  know  not  merely  the 
data  of  economics,  but  also  the  explanations  of  the  data 
which  the  science  of  economics  has  to  offer.  To  know 
history,  one  must  first  know  something  of  economics. 
To  be  an  historian,  one  must  first  be  something  of  an 
economist. 

In  so  far,  moreover,  as  history  sets  forth  foreign 
relations,  it  describes  public  activity  in  reference  to 
that  body  of  rules  which  are  supposed  to  govern 
nations  in  their  dealings  with  one  another ;  and  in 
order  to  understand  this  activity,  and  to  be  in  a  posi- 
tion to  speak  with  critical  judgment  upon  it,  which 
it  is  the  business  of  the  historian  and  the  teacher  of 
historj-  to  do,  one  must  know  this  body  of  rules;  in 
other  words,  one  must  have  a  sufficient  knowledge  of 
international  law. 

The  teacher  who  has  read  carefully  in  economics  and 
international  law  will  be  in  a  condition  to  bring  to  his 
classes  illustrations  and  explanations  which  hitherto 
have  been  generally  wanting  in  the  teaching  of  history, 
and  because  they  have  been  wanting  students  have 
regarded  the  subject  as  somewhat  stale,  flat,  and  un- 
profitable ;  and  in  this  matter  the  judgment  of  the 
students  does  not  appear  far  from  the  truth.  The 
history  of  the  action  of  the  French  Revolutionary 
government  regarding  its  revenues  and  expenditures, 
for  example,  has  no  interest,  and  appears  to  have  little 
significance,  except  as  we  are  able  to  bring  to  its  con- 
sideration a  knowledge  of  the  economics  of  money  and 
the  fundamental  principles  of  finance.  In  the  history 
of  this  country,  moreover,  there  are  many  phases  which 
cannot  be  fully  comprehended,  except  in  the  light  of 


Social  Sciences  in   Teaching  History.  23 

economic  principles ;  and  many  controversies  into 
which  we  have  fallen  with  other  nations  cannot  be 
adequately  appreciated  and  correctly  estimated,  ex- 
cept we  know  the  bearing  of  international  law  on 
the  questions  involved. 

But  as  an  aid  in  understanding  history,  the  science 
of  politics  is  even  more  important  than  economics  or 
international  law.  Dealing  with  the  nature,  functions, 
and  structure  of  the  state,  it  enables  one  to  get  clear 
and  definite  conceptions  of  the  elements  whose  activity 
constitutes  the  subject  of  the  historical  narrative  or 
discussion.  Unless  we  are  able  to  assign  ajjrecJM.- 
meaning  to  the  various  terms  used  relating  to  gov- 
ernment and  society,  historical  writing  will  neces- 
sarily appear  vague,  and  the  vagueness  will  either 
become  wearisome  and  prevent  long-continued  read- 
ing, or  will  engender  indifference  to  accuracy  of 
thinking.  But  if  we  know  that  sovereignty  means 
the  power  vested  somewhere  in  the  political  organism, 
by  which  the  fundamental  positive  law  is  issued  and 
may  be  changed,  and  that  the  sovereign  is  the  person 
or  organized  body  of  persons  holding  this  power,  we 
are  in  a  position  to  appreciate  the  full  significance  of 
these  terms  when  used  correctly,  and  to  detect  the  error 
when  used  incorrectly  ;  and  by  thus  always  moving  in 
the  light  of  a  clear  understanding,  we  may  avoid  con- 
fusion, and  hope  to  reach  the  truth.  On  nearly  every 
page  of  history  there  is  some  record  of  the  action  of 
the  state,  and  a  large  number  of  readers  are  in  the 
absurd  position  of  following  the  narrative  of  a  course 
of  activity,  without  having  a  clear  and  definite  con- 
ception of  the  limits  and  qualities  of  the  subject  acting. 
-  And  before  the  science  of  politics  was  developed  for 
the  enlightenment  of  the  historian,  his  work  showed  a 


24  Riverside  Addresses. 

tendency  to  become  a  royal  chronicle  amplified  by 
voluminous  details  of  military  affairs.  But  scientific 
analysis  of  political  phenomena  has  helped  to  clarify 
our  views  of  society,  and  make  history  in  this  age  more 
completely  than  ever  before  a  comprehensive  account 
of  the  progressive  national  life.  It  has  revealed  to  us, 
moreover,  the  distinction  between  the  state  and  the 
nation,  showing  the  state  not  as  a  divine  incompre- 
hensible something,  but  as  that  body  of  persons  in  the 
nation  who  have  political  rights,  and  are  united  in  an 
organization  to  wield  the  power  of  the  nation  in  behalf 
of  the  nation.  The  state,  therefore,  stands  revealed 
by  modern  political  science  as  a  purely  human  institu- 
tion, which  has  come  into  its  present  complex  form 
only  gradually  as  the  needs  of  public  agencies  to  do 
the  gradually  increasing  public  work  have  appeared. 

Through  the  science  of  politics  we  are  made  to  see 
not  only  what  the  state  is,  that  is  to  say,  what  the 
organism  is  whose  actions  form  a  large  part  of  the 
subject-matter  of  history,  but  also  to  see  by  what  special 
agencies  within  the  state  the  various  departments  of 
the  public  work  of  society  are  carried  on.  On  this  point 
the  student  in  the  early  }Tears  of  his  study  of  history 
has  much  need  of  enlightenment ;  for  the  historian, 
while  treating  of  the  progress  effected  by  the  public 
action  which  has  been  carried  on  through  these  agencies, 
often  leaves  us  entirely  uninformed  as  to  the  nature, 
structure,  and  relations  among  themselves  of  these 
agencies.  Yet,  under  the  present  organization  of 
instruction,  students  ma}r  not  be  expected  to  acquire 
this  information  before  beginning  the  study  of  history, 
however  necessary  it  may  be  to  clearness  and  accuracy 
of  historical  knowledge. 

If,  then,  we  conclude  that  much  of  the  information 


Social  Sciences  in   Teaching  History.  25 

which  is  involved  in  -the  sciences  of  economics  and 
politics  is  essential  to  an  enlightened  understanding  of 
history,  and  that  students  cannot  be  required  to  make 
independent  acquisitions  in  this  line  until  late  in  their 
academic  life,  we  are  obliged  to  find  means  to  remedy 
this  defect  and  prevent  their  early-acquired  historical 
knowledge  from  being  vague  and  unprofitable  ;  and  in 
this  search  we  are  turned  to  the  method  of  teaching. 

In  presenting  a  subject  in  history,  no  teacher  in  these 
days  is  likely  to  be  satisfied  to  leave  the  topic  with  the 
meager  statement  of  the  text-book  ;  and  he  is,  there- 
fore, compelled  to  devise  some  feasible  means  for  ex- 
tending this  information.  In  this  matter  he  will  be 
led  to  accept  not  necessarily  the  best  means  abstractly 
considered,  but  the  most  serviceable  in  view  of  the 
time  and  appliances  at  command.  These  usually  reduce 
themselves  to  two.  The  first  is  to  require  the  student 
to  read  indicated  passages  in  various  authors  on  the 
topics  treated  in  the  text.  Given  practically  unlimited 
time,  all  the  books  desired,  and  a  certain  ripeness  of 
the  critical  judgment,  this  method  may  be  employed 
with  great  advantage.  But  these  conditions  are  all 
wanting  in  the  schools  below  the  university,  and  even 
in  the  lower  classes  of  the  university  itself.  When- 
ever a  large  class  is  being  taken  over  a  given  course  of 
instruction,  and  all  the  members  are  reading  the  same 
list  of  indicated  passages,  they  will  meet  with  embar- 
rassment in  any  library  in  this  country,  on  account  of 
lack  of  books.  This  plan  is,  therefore,  practicable  only 
after  the  elementary  stages  of  historical  instruction  have 
been  passed,  those  stages  in  which  the  work  of  all  the 
students  is  of  necessity  essentially  the  same. 

Even  if  these  physical  difficulties  were  removed,  it 
would   not   be   advisable   to  give   this   method   more 


26  Riverside  Addresses. 

than  a  limited  application.  In  the  early  days  of  the 
discovery  of  German  universities  by  American  students, 
the  methods  there  employed  made  a  profound  impres- 
sion. The  young  men  who  came  back  to  teach  in 
American  schools  or  colleges,  were  moved  to  give 
instruction  by  lectures.  They  forgot  the  conspicuous 
differences  between  their  own  positions  and  the  posi- 
tion of  the  German  professor.  They  forgot  that  the 
German  students  came  to  the  lectures  already  possess- 
ing many  of  the  attainments  which  belong  to  a  liberal 
education,  and  that  the  professors  had  something  to 
communicate  not  found  in  the  ordinary  text-book; 
often  the  results  of  their  own  investigations,  which 
had  not  been  embodied  in  the  literature  of  the  subject 
in  hand.  The  professor  of  the  German  university 
spoke  to  young  men  already  familiar  with  the  elements 
and  outlines  of  the  sciences,  and  who  wished  to  learn 
the  sources  of  further  information.  Pointing  out  and 
criticising  these  sources  was  often  a  large  part  of  the 
lecturer's  work.  These  conditions  were  all  wanting, 
and  are  still  wanting,  in  a  large  part  of  the  field  of 
academic  instruction  in  this  country;  and  the  German 
method  of  lecturing,  apart  from  its  proper  conditions, 
was  absurd.  It  had  an  application  and  was  the  only 
practicable  method  in  the  early  history  of  education, 
when  books  were  few  and  inaccessible  to  most  learners. 
Again,  in  these  times,  when  books  have  become  bewil- 
deringly  abundant,  the  academic  lecture,  in  so  far  as  it 
is  a  giving  of  special  methods  of  investigation  and 
criticism  of  sources,  has  become  useful  for  students  of 
large  attainments.  These  conditions  may  be  found  in 
the  graduate  department  of  our  universities,  and  per- 
haps to  a  certain  extent  in  the  highest  grades  of  the 
undergraduate  courses. 


Social  Sciences  in   Teaching  History.  27 

In  these  last  years,  moreover,  the  American  teacher 
has  discovered  the  seminary  method.  This  was  also 
brought  to  light  in  Germany.  The  plan  of  instruction 
which  it  involves  has  numerous  excellences,  but  it  is 
not  adapted  to  all  circumstances.  It  is  undoubtedly  a 
proper  auxiliary  in  the  highest  grades  of  instruction 
where  students  under  competent  direction  are  pursu- 
ing independent  lines  of  research.  But  its  extensive 
employment  in  the  early  years  of  historical  study  is 
questionable.  Specialized  investigations  in  history 
from  the  beginning  of  the  student's  career  lead  to  the 
danger  of  knowing  everything  without  knowing  the 
whole.  The  knowledge  of  the  facts  relating  to  any 
particular  event,  unsupported  by  a  synthetic  under- 
standing of  the  whole  course  of  human  experience,  and 
by  an  appreciation  of  the  historic  place  of  this  event, 
is  scarcely  more  important  than  a  knowledge  of  the 
fictitious  events  and  characters  of  Scott  or  Balzac. 
History  as  an  academic  study  means  more  than  this; 
and,  to  make  it  more,  some  plan  of  study  must  be 
had  which  shall  defer  the  minute  investigation  of 
special  points  till  the  mind  has  been  trained  to  grasp 
as  a  unit  the  whole  record  of  progressive  society.  If 
one  would  be  an  historian,  or  merit  mastership  in  this 
department  of  knowledge,  it  must  become  habitual  for 
him  to  view  events  in  the  large.  To  this  end,  the 
scheme  of  western  history  should  be  unfolded  before 
him  at  the  beginning  of  his  studies,  and  kept  before 
him  through  all  his  subsequent  work.  At  first,  a  mere 
skeleton  in  his  mind,  from  year  to  year  it  acquires  a 
greater  and  greater  fullness  of  detail,  until  at  last  he 
finds  himself  in  possession  of  a  comprehensive  and 
well-balanced  general  knowledge  of  history.  At  this 
point  some  limited  field  may  be  selected,  and  made  the 


28  Riverside  Addresses. 

object  of  his  special  inquiries.  To  this  he  ma}T  then 
direct  whatever  force  and  scholarly  insight  he  can  com- 
mand, and  not  be  in  danger  of  failing  to  appreciate  the 
proper  relation  of  his  special  subject  to  the  whole  of 
which  it  is  a  part.  But  to  direct  the  immature  student 
to  the  stud}'  of  special  points  in  history  is  to  cause  him 
to  become  habituated  to  small  views,  and  to  render  him 
liable  to  be  hopelessly  lost  in  details.  He  may  know 
the  record  of  certain  events,  but  never  rise  to  the  larger 
conception  of  history. 

The  second  plan  for  extending  the  range  of  the  stu- 
dent's information  is  that  under  which  the  teacher  fur- 
nishes a  more  or  less  elaborate  commentary  on  the  points 
involved  in  the  text.  In  carrying  out  this  plan,  an  item 
of  first  importance  is  to  determine  what  books  shall  be 
used  by  the  students.  An  objection  is  raised  against 
the  use  of  the  ordinary  text-book,  on  the  ground  that 
it  is  meager  and  lifeless,  and  that  supplementary  read- 
ing cannot  be  done  by  a  large  class  with  advantage.  If 
there  were  no  physical  difficulties  in  the  way,  students 
would  be  likely  to  read  extensively  on  some  points,  and 
only  a  little  on  other  points,  perhaps  more  important, 
according  to  the  facility  of  obtaining  the  literature 
desired.  Both  of  these  difficulties  may  be  avoided  by 
putting  into  the  hands  of  each  student  a  book  by  a 
recognized  master  of  the  subject  to  be  considered,  and 
not  by  a  mere  text-book  maker,  which  shall  contain  all 
that  we  may  require  the  student  to  read,  even  when 
under  the  highest  pressure.  If,  to  illustrate  by  a  single 
example,  the  subject  for  the  term  in  the  university  is 
English  history  in  the  eighteenth  century,  let  each 
student  have  the  eight  volumes  of  Lecky's  "  History  of 
England  in  the  Eighteenth  Century."  The  advantages 
of  this  over  the  small  text-book  and  outside  reading 


Social  Sciences  in   Teaching  History.  29 

are  numerous.  In  the  first  place,  what  the  student 
reads  on  the  subject  will  be  in  proportion  to  the  impor- 
tance of  the  several  parts  of  it,  since  he  will  take  it  in 
the  form  in  which  a  great  historian  has  viewed  it  and 
set  it  forth.  He  will,  moreover,  have  all  the  material 
of  his  work  at  hand  for  reference  at  all  times.  He  will- 
learn  to  read  a  voluminous  book,  which  is  a  matter  of 
vast  importance  for  the  youth  who  would  be  a  student ; 
and  after  a  few  undertakings  of  this  kind,  he  will  not 
be  deterred  from  seeking  information  because  it  happens 
to  be  contained  in  a  work  of  several  volumes.  The  diffi- 
culties which  arise  from  attempts  to  supplement  the 
deficiencies  of  the  small  text-book  do  not  appear  here  ; 
and  if  the  opposite  may  sometimes  hold,  the  teacher 
may  use  his  discretion  in  directing  limitations,  and  no 
embarrassment  will  ensue.  With  such  a  text  before 
the  student,  the  teacher  will  not  be  tempted  to  waste 
time  in  giving,  by  way  of  informal  lectures,  what  can 
be  much  better  acquired  from  the  book  itself.  Never- 
theless there  should  be  comment,  but  the  nature  of  it 
will  depend  very  largely  on  the  character  of  the  book 
used ;  but  in  general  it  may  be  said  that  the  teacher 
should  illustrate  by  setting  forth  the  principles  of  social 
order  and  progress,  rather  than  consume  the  time  in 
elaborating  the  narrative.  Merely  extending  the  story 
of  historical  events  adds  very  little  to  our  understand- 
ing of  the  subject ;  but  analysis  of  the  state  and  society 
reveals  to  us  the  mode  of  organization,  the  instrumen- 
talities of  social  action,  and  the  forces  by  which  the 
nation  is  moved  along  the  course  of  improvement. 

But  this  involves  on  the  part  of  the  teacher  certain 
attainments  in  economics  and  political  science,  and  it  is 
the  importance  of  these  attainments  for  the  teacher  of 
history  that  I  wish  to  emphasize.     The  special  char- 


30  Riverside  Addresses. 

acter  which  historical  studies  will  assume  under  differ- 
ent circumstances  will  depend  very  much  on  the  kind 
of  opportunities  which  .  these  different  circumstances 
present.  In  'European  countries  where  the  scholar  has 
access  to  the  sources  of  history,  whether  in  great  libra- 
ries or  in  the  archives,  his  investigations  are  likely  to 
be  largely  directed  to  setting  forth  hitherto  unknown 
details.  But  in  a  country  like  this,  whose  civilization 
is  but  of  yesterday,  and  which  has  no  sources  except 
for  the  history  of  a  few  recent  generations,  historical 
study  will  necessarily  be  turned  to  some  other  end. 
The  American  scholar  is  unable  to  compete  success- 
fully with  the  European  scholar  in  searching  out  the 
details  of  early  European  history;  and  his  energy  and 
taste  for  social  studies  will  tend,  therefore,  to  find  satis- 
faction in  the  scientific  interpretation  of  the  historical 
facts  presented  by  European  historians,  and  in  the 
development  of  those  social  sciences  which  take  their 
data  from  history. 

But  hitherto  the  achievements  of  this  nation  on  any 
of  these  lines  have  not  been  remarkable,  yet  this  fact 
does  not  indicate  a  lack  of  worthy  political  thought. 
In  the  field  of  political  theory,  Roman  literature,  to 
illustrate,  is  specially  barren,  yet  no  one  will  deny  that 
the  Romans  thought  much  and  well  on  political  prob- 
lems. Their  thinking  was  given  expression  and  form 
inlaws  and  practical  institutions  rather  than  in  treat- 
ises; and  in  this  regard  the  people  of  the  United  States 
are  comparable  with  the  Romans.  Eor  more  than  a 
hundred  years  this  nation  has  been  deeply  absorbed  in 
the  work  of  building  a  great  state;  and  the  best  the 
nation  has  thought  has  been  embodied  in  the  institu- 
tions and  laws,  giving  them  in  some  sense  the  char- 
acter of  original  creations.    But  the  large  place  occupied 


Social  Sciences  in   Teaching  History.  31 

by  social  subjects  in  the  universities,  and  the  large 
number  of  men  giving  to  these  subjects  their  thought- 
ful attention,  are  conditions  specially  favorable  to  the 
production  of  an  important  economic  and  political 
literature.  Here  the  parallel  with  Rome  ends.  While 
the  Roman  republic  fell  under  imperial  rule  which 
choked  all  political  discussion,  this  republic  has  entered 
upon  that  phase  of  its  history  in  which  thoughtful 
political  discussion  has  become  a  necessary  condition  of 
continued  healthy  existence.  And  the  present  activity 
in  the  study  of  society  suggests  that  the  nation  may  yet 
make  achievements  in  economics  and  political  science 
worthy  of  the  originality  already  displayed  in  practical 
politics. 

There  are  good  reasons  why  some  part  in  cultivating 
this  field  should  fall  to  the  teachers  of  history,  not 
merely  in  the  universities  but  also  in  the  other  schools 
throughout  the  country.  The  first  of  these  is  the  gen- 
eral fact  that  independent  work,  which  results  in  an 
advancement  of  knowledge,  is  essential  to  the  highest 
efficiency  of  the  teacher.  The  teacher  who  ceases  to 
be  an  investigator,  who  ceases  to  seek  to  enlarge  his 
mental  vision  by  scholarly  inquiries,  is  in  danger  of 
stagnation  and  the  consequent  loss  of  the  reasonable 
ground  of  his  calling.  Another  circumstance  by  which 
the  already  mentioned  line  of  study  is  commended 
as  a  proper  field  for  the  teacher's  independent  inquiries, 
is  the  fact  that,  while  in  the  realm  of  pure  historical 
research  adequate  sources  of  information  are  found  in 
only  a  few  places,  in  the  auxiliary  subjects  of  economics 
and  political  science  one  may  always  have  at  hand 
abundant  material  for  original  work.  There  are  always 
within  his  horizon  the  data  of  some  hitherto  unsolved 
social  problem.     The  society  in  which  he  lives  is  the 


32  Riverside  Addresses. 

laboratory  in  which  experiments  are  constantly  going 
on  before  his  eyes. 

The  analysis  of  social  institutions,  moreover,  will 
facilitate  an  understanding  of  that  peculiar  and  distinct- 
ive characteristic  of  western  society  which  we  call 
progress,  and  thus  throw  a  light  on  history  that  cannot 
be  derived  from  any  other  source.  In  this  light  one 
may  see  clearly  the  unsoundness  of  the  favorite  expres- 
sion of  the  half-instructed,  that  history  repeats  itself. 
With  our  knowledge  of  history,  enlightened  by  an 
understanding  of  the  structure  and  moving  forces  of 
society,  it  becomes  evident  that  that  movement  which 
is  observed  in  progressive  society  is  never  in  a  circle, 
but  always  on  to  new  ground,  though  not  always  in 
an  unswerving  line  forward.  Progress  is  not  contin- 
uous, nor  always  at  the  same  rate;  there  are  periods  of 
halting,  and  even  of  retrogression,  when  deterioration 
marks  many  of  the  phases  of  life,  when  a  mortal  disease 
appears  to  have  smitten  a  whole  civilization.  It  per- 
ishes like  the  seed  that  is  sown,  but  after  the  destruc- 
tion there  rises  a  new  and  more  fruitful  life.  The 
growth  of  society  is,  moreover,  like  the  growth  of  a 
great  city;  built  up  little  by  little,  it  serves  the  pur- 
poses of  a  generation,  but  at  last  fails  to  meet  the  new 
needs.  One  by  one  its  buildings  are  replaced  by  others 
of  better  plan  and  sounder  make.  Thus  in  social 
growth,  old  institutions  that  have  fulfilled  their  mis- 
sion yield  in  time  to  institutions  suited  to  the  new 
wants  of  a  more  advanced  society.  A  great  revolution- 
ary movement  sometimes  sweeps  the  institutions  of  a 
nation  to  destruction  in  a  day,  as  a  fire  sweeps  away 
the  various  structures  that  have  served  the  diverse 
purposes  of  civilized  existence.  Then  the  work  of 
construction  is  rapid  and  free;  and  as  the  new  city  is 


Social  Sciences  in   Teaching  History.  33 

better  than  the  old,  so-are.  the  ae^v  institutions -adapted, 
to  a  higher  and  better  social  existence.  But  the  old 
structures  are  never  rebuilt,  nor  the  old  institutions 
recalled  into  active  force. 

This  general  truth  finds  another  expression  in  the 
statement  of  the  fact  that  the  practical  social  problems 
of  no  two  ages  are  ever  exactly  the  same.  If  the  ages 
are  far  apart,  all  the  conditions,  even  the  nature  of  the 
persons  involved,  may  be  different.  The  men  of  this 
age,  for  example,  who  know  from  observation  the 
marvels  of  modern  life,  have,  as  the  creatures  of  other 
circumstances  and  other  traditions,  a  nature  different 
from  that  of  the  men  who  lived  the  narrow  human 
existence  of  the  Middle  Ages.  They  are  distinguished 
by  a  wider  range  of  intellectual  vision,  a  more  compre- 
hensive grasp  of  the  imagination,  and  a  series  of  new 
impulses.  Moreover,  the  physical  conditions  of  to-day 
are  utterly  at  variance  with  those  of  earlier  centuries; 
and  these  two  facts  alone  are  adequate  to  make  the 
social  problems  which  we  have  to  solve  new  problems. 
Because  the  problems  of  each  age  are  new,  social 
advancement  can  be  achieved  only  by  slow  and  costly 
experiments.  The  general  place  of  experiment  in  the 
progress  of  society  the  teacher  of  history  should  under- 
stand, and  for  enlightenment  he  must  turn  to  those 
writers  whose  business  it  is  to  analyze  society  and  to 
set  forth  the  laws  of  social  order  and  growth. 

The  fact  that  some  points  are  still  in  debate  in  the 
sciences  which  deal  with  the  phenomena  of  society,  is 
not  an  argument  against  their  utility  as  subjects  of 
study.  Every  science  has  a  margin  of  unsettled  ques- 
tions. These  constitute  the  principal  attraction  for  the 
independent  investigator,  and  furnish  him  an  oppor- 
tunity to  make  a  permanent  contribution  to  human 

3— A 


34  Riverside  Addresses. 

knowledge.  The  comparatively  late  development  of 
the  social  sciences  is  not  wholly  due  to  the  influence 
of  absolute  government  practically  prohibiting  the 
discussion  of  questions  relating  to  political  affairs,  but 
also  in  part  to  the  complexity  of  the  data  involved. 
The  attempts  to  render  a  scientific  account  of  society 
have  made  unusually  prominent  certain  social  phe- 
nomena, which,  since  they  have  been  made  specialty 
conspicuous,  the  teacher  of  history  is  bound  to  consider. 
Even  if  he  would,  he  may  not  present  history  as  a 
chronicle  of  the  achievements  of  the  distinguished  head 
of  the  state.  He  must  set  it  forth  in  its  character  as 
the  record  of  the  most  complex  of  organisms ;  and  to 
learn  this  character  there  is  no  better  source  than  the 
writings  of  those  by  whom  this  complexity  has  been 
specially  revealed,  and  through  whom  a  certain  degree 
of  scientific  order  has  been  brought  out  of  the  chaos  of 
social  phenomena. 

But  political  science  has  encountered  even  more 
hinderances  than  economics.  The  claims  of  monarch- 
ical rulers  to  hold  a  power  of  divine  origin,  and  to 
be  free  from  responsibility  to  the  governed,  made  the 
theory  of  politics  in  some  sense  an  appendage  of  theel- 
ogy.  From  the  conception  of  the  monarch  as  the  agent 
of  God,  and  of  his  power  as  the  supreme  power  in  the 
state,  the  step  was  easy  to  the  conclusion  that  the  state 
itself  was  divine.  Metaphysics  applied  to  politics  be- 
came sterile  and  unprofitable.  The  only  possibility  of 
progress  lay  in  taking  a  new  point  of  view  and  a  new 
method.  This  was  not  readily  accomplished,  for  men 
hesitated  to  encounter  the  thunders  of  the  metaphysical 
theologians,  or  the  theological  metaphysicians,  which 
would  necessarily  follow  the  declaration  that  their 
whole  philosophy  of  the  state  was  unreal  and  empty. 


Social  Sciences  in   Teaching  History.  35 

Nevertheless,  the  interests  of  progressive  knowledge 
demanded  that  the  assumptions  of  metaphysical  politics 
should  be  abandoned,  and  a  new  start  made  from  ob- 
served and  known  data.  With  this  beginning,  and 
using  the  methods  of  ordinary  scientific  research,  im- 
portant steps  have  already  been  taken  toward  the 
creation  of  a  science  of  politics  that  will  explain  the 
phenomena  of  which  it  treats.  And  it  is  to  this,  and 
not  to  the  theologico-metaphysical  politics,  that  the 
teacher  must  look  for  light  on  the  political  and  consti- 
tutional phases  of  history. 

An  important  function  of  the  teacher  of  history, 
therefore,  under  the  method  suggested  for  work  below 
the  highest,  is  that  of  a  judicious  commentator  on  the 
topics  involved  in  the  student's  reading,  and  in  making 
these  comments  no  information  will  serve  his  purposes 
better  than  that  drawn  from  economic  and  political 
science,  in  which  are  set  forth  the  laws  of  social  organ- 
ization as  well  as  the  forces  and  methods  of  progress. 


Riverside  Addresses. 


THE  PAST  AND  PRESENT  OF  ELEMENTARY 
MATHEMATICS. 


By  Irving  STringham, 

Professor  of  Mathematics. 


W  In  the  long  history  of  the  world's  intellectual  progress 
nothing  is  more  striking  than  the  dependence  of  each 
succeeding  age  upon  its  predecessors  for  the  materials 
out  of  which  its  own  achievements  are  wrought.  The 
raw  materials  we  use  in  the  manufacture  of  science  are 
seldom  dug  up  fresh  from  the  earth,  but  are  picked  up 
by  the  wayside,  where  they  were  dropped  by  our  prede- 
cessors, who  could  devise  no  use  for  them.  It  thus 
happens  that  the  discoveries  of  Newton  are  old  hints 
wrought  over  and  made  into  the  finished  product  of 
science  by  the  master-mind,  some  of  them  dating  back- 
wards nineteen  centuries,  to  the  time  of  Archimedes. 
The  most  transcendant  genius  works  under  limitations, 
cannot  see  far  into  the  future ;  and  if,  by  a  fortuitous 
inspiration  he  anticipates  a  discovery  that,  standing  out 
of  relation  to  the  knowledge  of  his  own  time,  belongs 
to  a  succeeding  age,  it  remains  unused,  unfruitful,  and 
of  no  current  value  till  such  time  as  it  can  be  correlated 
with  other  scientific  knowledge. 

And  so  every  science  has  a  prehistoric  stage,  in 
which  principles  prospectively  important  are  at  first 
merely  the  incidents  in  apparently  unrelated  problems 
or  investigations,  their  first  users  perhaps  entirely 
unknown   and   never   to   be   identified.      Only   later, 


Elementary  MatJiematics.  37 

possibly  after  repeated  recurrence  in  investigations  of 
which  it  is  the  true  foundation,  does  the  principle  call 
attention  to  itself  as  of  great  importance.  It  is  the 
recognition  of  its  importance,  not  its  first  merely  inci- 
dental use,  that  constitutes  real  discovery. 

The  prehistoric  period  of  mathematics  belongs  to  the 
centuries  immediately  preceding  the  earliest  develop- 
ment of  Greek  philosophy,  and  appears  to  have  been 
first  cultivated  in  connection  with  land  surveying  and 
astronomy  in  Egypt  and  Assyria.  It  had  its  begin- 
nings as  a  science  in  the  latter  part  of  the  seventh 
century,  B.  C,  at  Miletus,  where  the  Greek  philosopher 
Thales,  who  had  traveled  extensively  in  foreign  coun- 
tries and  had  resided  for  a  time  in  Egypt,  first  taught 
geometry  deductively;  but  nearly  contemporaneously 
also  at  Crotona,  where  Pythagoras  first  established  the 
principles  of  the  doctrine  of  proportion,  and  laid  the 
foundations  of  a  goodly  part  of  elementary  geometry. 
Thenceforward,  so  long  as  Greek  civilization  existed 
anywhere,  the  study  of  geometry  never  ceased  to  be  an 
important  factor  in  the  world's  intellectual  progress, 
and  for  more  than  twelve  centuries  the  continuity  of 
its  study  was  unbroken.  First,  Thales  at  Miletus, 
then  Pythagoras  at  Crotona,  then  Hippocrates  at 
Athens,  then  Euclid  at  Alexandria,  then  Archimedes 
at  Syracuse,  then  Apollonius  at  Perga,  then  Pappus  at 
Alexandria,  wrought  out  in  succession,  classified  and 
organized  into  scientific  unity  the  great  mass  of  propo- 
sitions which  constitute  the  subject-matter  of  geomet- 
rical study  in  our  schools  to-day. 

Near  the  close  of  the  Greek  period,  during  which 
geometry,  from  the  merest  beginnings,  had  grown  into 
a  comprehensive  system,  the  first  published  accounts 
of  an   algebraic   analysis    appear,  the   Arithmetica  of 


38  Riverside  Addresses. 

Diophantos,  the  last  contribution  of  the  great  Alexan- 
drian school  to  mathematical  science.  This  work, 
remarkable  for  its  achievement  of  results,  but  lament- 
ably defective  in  method  and  organic  unity,  had  in 
it  the  foreshadowings  of  a  new  science,  but  was  in  no 
respect  the  science  itself.  It  possessed  no  power  of 
further  growth  from  within,  and  when  algebra  finally 
became  scientific  it  was  constructed  on  entirely  new 
models. 

But  even  if  the  work  of  Diophantos  had  been  con- 
structed upon  correct  scientific  principles,  it  would  still 
have  passed  speedily  into  oblivion,  for  the  light  of 
Greek  civilization  was  already  feeble  and  flickering 
and  was  soon  completely  extinguished  by  the  fanati- 
cism and  bigotry  that  for  seven  centuries  enveloped 
Europe  in  almost  total  darkness.  In  fact,  the  work  of 
Diophantos  was  not  rediscovered  in  Europe  until  the 
middle  of  the  sixteenth  century. 

With  the  disappearance  of  the  Greek  schools  there 
seemed  to  be  no  hope  for  the  further  cultivation  of 
mathematics  as  a  science  on  the  face  of  the  earth,  for 
nowhere  else  in  the  ancient  world,  up  to  that  time,  had 
any  results  of  a  high  order  been  achieved.  But  at  this 
critical  juncture  a  new  light  appears  for  us  in  the  east. 
The  Arithmetica  of  Diophantos  had  been  published 
(conjecturally)  in  the  fourth  century  of  our  era,  and  the 
Alexandrian  school  continued  in  existence  until  the 
Mohammedan  conquest  in  641  A.  D.,  during  which 
time  mathematics  was  still  cultivated,  though  feebly, 
in  the  form  of  commentary  or  perfunctory  study,  and 
without  originality  or  fruitful  result ;  and  it  was  more 
than  a  century  and  a  half  before  the  final  catastrophe — 
the  capture  of  Alexandria  and  the  burning  of  the  great 
library  by  the  Mohammedans — that  there  appeared  in 


Elementary  Mathematics.  39 

India  a  work  on  algebra  and  trigonometry  by  the 
astronomer  Aryabhatta,  of  which  no  Greek  mathema- 
tician of  the  earlier  centuries  would  have  been  capable. 
From  this  time  until  the  revival  of  learning  in  western 
Europe  the  Indians  are  the  true  discoverers  in  mathe- 
matical science.  To  what  extent  they  were  indebted 
to  the  Greeks  for  the  raw  materials  out  of  which  they 
constructed  their  algebra  is  not  known,  nor  is  it  of 
consequence,  for  since  the  Greeks  never  succeeded  in 
constructing  an  organic  system  of  algebra,  the  indebt- 
edness could,  in  any  event,  have  been  but  small. 

To  the  Indians,  then,  is  due  the  credit  of  first  creating 
algebra  as  a  science.  Two  great  works  on  mathematics 
and  astronomy  attest  this  claim  ;  one  by  Brahmagupta, 
written  in  628,  which  expounds  a  complete  system  of 
algebraic  analysis,  the  other  by  Bhaskara,  in  1 1 50,  upon 
arithmetic  and  algebra,  in  which  the  Indian  system  of 
arithmetic — the  one  we  ourselves  use — is  employed. 
These  works,  however,  record  the  highest  achievements 
of  the  Indians  in  mathematics,  and  all  subsequent  prog- 
ress in  algebra  has  taken  place  in  the  west,  and  in 
modern  times. 

Though  the  Indians  had  shown  themselves  to  be 
consummate  algebraists,  they  were  in  no  sense  geome- 
ters; and  far  from  adding  any  new  thought  to  the  sci- 
ence which  the  Greeks  had  created,  they  apprehended 
with  difficulty  and  with  much  blundering  what  of  geom- 
etry they  received  from  Greek  sources.  Thus  two  great 
civilizations  had  stood  on  either  side  of  a  barrier  which 
neither  could  pass.  The  Greeks  had  created  geometry, 
but  they  could  not  invent  algebra;  the  Indians  invented 
algebra,  but  they  could  not  add  one  proposition  of  im- 
portance to  geometry. 

Out  of  these  two  independent  sources  issued  the  two 


40  Riverside  Addresses. 

distinct  streams  of  mathematical  thought  that  flowed 
first  sluggishly  through  the  Mohammedan  countries  of 
Arabia,  Africa,  and  Spain,  thence  finally  into  Christian 
Europe,  where  they  were  subsequently  joined  together. 
For  it  was  from  the  Arabs  that  mediaeval  Europe  first 
acquired  in  the  twelfth  century  some  knowledge  of  both 
algebra  and  geometry.  The  Arabs,  however,  were  not 
good  conservers,  or  compilers,  of  the  scientific  knowl- 
edge accessible  to  them  from  Greek  and  Indian  sources, 
and  transmitted  it  in  such  imperfect  form  that  many 
important  principles  had  either  to  be  rediscovered,  or 
sought  for  in  the  Greek  or  Sanskrit,  before  Europe 
acquired  full  possession  of  the  mathematical  knowl- 
edge which  Greece  on  the  one  hand  and  India  on  the 
other  had  contained.  But  be  this  as  it  may,  Europe 
rapidly  recovered,  during  the  four  centuries  from  the 
twelfth  to  the  sixteenth,  substantially  all  the  mathe- 
matical knowledge  that  the  ancient  civilizations  had 
bequeathed  to  their  successors. 

During  the  sixteenth  century  mathematics  received 
its  share  of  that  activity  in  intellectual  pursuits  which 
has  received  the  name  of  the  Renaissance;  but  its 
development  was  strictly  upon  the  lines  that  had  been 
marked  out  for  it  in  ancient  times.  Geometry  did  not 
free  itself  from  the  limitations  as  to  method  and  scope 
that  had  been  set  for  it  by  Euclid,  Archimedes,  and 
Apollonius.  Algebra  followed  the  Indian  model,  which 
had  been  recovered  with  much  difficulty  by  the  help  of 
the  Arabs,  and,  barring  a  few  geometrical  constructions 
of  algebraic  equations,  mere  .  translations  into  modern 
forms  of  expression  of  well-known  Greek  demonstra- 
tions, did  not  get  beyond  the  limited  definition  of 
algebraic  quantity  as  rational  or  commensurable  num- 
ber.    Algebra  and  geometry  were  still   two   distinct 


Elementary  Mathematics.  41 

sciences,  and  there  were  no  indications,  except  in  a  few 
special  instances,  that  the  one  was  in  any  way  suscepti- 
ble of  interpretation  in  terms  of  the  other. 

In  the  seventeenth  century,  however,  three  important 
steps  were  taken  towards  a  recognition  of  the  intimate 
relations  that  we  now  know  these  two  main  divisions 
of  elementary  mathematics  bear  to  each  other,  and 
which  play  such  an  important  part  in  all  modern  math- 
ematical investigations.  The  first  was  through  the 
invention  of  logarithms,  in  1 614,  by  Napier;  the  second 
through  the  invention  of  analytic  geometry,  in  1637, 
by  Descartes;  the  third  through  the  invention  of  the 
differential  calculus,  in  1665,  by  Newton. 

The  introduction  of  logarithms,  however,  important 
as  it  was  as  an  essential  part  of  the  new  analysis  soon 
to  be  created,  was  recognized  at  this  time  only  as  sim- 
plifying the  processes  of  arithmetical  calculation. 

Both  the  analytic  geometry  and  the  differential  cal- 
culus seemed  to  be  necessary  prerequisites  for  a  full 
understanding  of  logarithms  as  a  part  of  analysis,  and 
a  century  passed  before  their  importance  as  such  was 
recognized;  furnishing,  in  fact,  a  remarkable  instance 
of  a  premature  discovery.  The  ability  to  recognize  the 
true  import  of  the  logarithm  was  made  possible  only  by 
discoveries  that  came  half  a  century  after  the  logarithm 
itself  had  been  invented. 

The  discovery  made  known  to  the  world  by  Des- 
cartes, in  1637,  may  be  summed  up  in  the  words:  Every 
function  has  a  graph.  To  illustrate  by  a  very  simple 
example,  the  function  x'1  has  for  its  graph  a  parabola 
with  its  principal  vertex  at  the  origin  of  coordinates, 
and  its  principal  diameter  coincident  with  the  j-axis. 
Thus: 


42 


R iverside  A  ddr esses. 


The  values  of  x  are  represented  by  lines  drawn  hori- 
zontally from  the  origin  O,  to  the  left  or  right,  from  the 
extremities  of  which  lines  are  drawn  upwards  to  repre- 
sent the  values  of  x1.  Each  pair  of  lines  thus  deter- 
mined by  a  pair  of  values  x,  x\  fixes  a  point  in  the 
plane,  .and  the  aggregate  of  all  such  points  range  them- 
selves along  the  curve  known  as  the  parabola.  The 
second  quantity  x1,  it  is  now  customary  to  represent  by 
a  second  letter,  as  y,  and  to  speak  of  x  and  y  as  func- 
tions of  each  other.  In  a  similar  manner  any  algebraic 
equation  between  two  quantities,  x  andjy,  has  its  graph. 

Now  mark  the  radical  departure  here  taken  in  the 
interpretation  of  algebraic  quantities.  A  straight  line 
stands  as  the  representative  of  any  such  quantity,  an 
interpretation  wholly  repugnant  to  Greek  geometry. 
So  long  as  his  practice  conformed  to  the  canon  which, 
from  time  immemorial,  had  been  his  guide  in  the  geo- 
metrical interpretation  of  quantity,  x'1  could  mean  for 
the  Greek  only  a  square,  an  area,  never  a  straight 
line.  But  his  orthodoxy  was  the  barrier  to  his  further 
progress,  and  from  the  abandonment  of  that  orthodoxy 
the  modern  mathematician  dates  the  possibility  of 
achievement  beyond  the  limits  of  investigation  which 
the  Greeks  had  set  for  themselves.     Henceforth,  more- 


Elementary  Mathematics.  43 

over,  the  interests  of  algebra  and  geometry  were  one 
and  the  same;  progress  in  the  one  was  to  mean  a  simul- 
taneous progress  in  the  other. 

Twenty-eight  years  pass,  and  we  stand  at  the  thresh- 
old of  the  greatest  of  the  discoveries  in  mathematics 
of  modern  times — that  of  the  differential  calculus.  The 
year  is  1665,  when  Sir  Isaac  Newton  communicated  to 
some  of  his  friends  the  fundamental  ideas  of  the  new 
method. 

Taken  in  connection  with  the  new  interpretation  of 
the  algebraic  equation  by  Descartes,  the  scope  of  this 
method  was  so  great  as  eventually — that  is,  within 
another  century  at  most — to  reconstitute  the  entire 
body  of  mathematical  science  upon  a  new  basis,  and 
to  completely  change  the  attitude  of  mathematicians 
towards  the  problem  of  its  further  advancement. 
Henceforth  its  various  parts  are  not  looked  upon  as 
disassociated  systems  having  no  common  meeting 
ground,  but  geometry,  algebra,  trigonometry,  analytic 
geometry,  the  differential  and  integral  calculus,  are 
seen  to  constitute  one  organic  whole,  or  rather  one 
organic  unit  in  a  much  larger  whole.  Henceforth 
mathematics  thus  newly  constituted  is  to  have  its 
alphabet,  its  nomenclature,  its  language,  and  whoever 
would  use  it  for  any  purpose  must  learn  its  nomencla- 
ture and  speak  its  language.  Henceforth  it  is  not  to 
be  the  plaything  of  the  philosopher,  nor  the  recreative 
and  disciplinary  study  of  the  scholiast,  but  the  instru- 
ment of  research  and  of  efficient  accomplishment  of 
practical  ends  in  the  hands  of  the  astronomer,  the  phys- 
icist, and  the  engineer.  Such  was  the  unparalleled 
achievement  of  the  seventeenth  century. 

But  I  must  be  more  explicit.     I  am  endeavoring  to 
trace  out,  in  brief  outline,  from  the  earliest  times  to  the 


44  Riverside  Addresses. 

present  da}',  the  development  of  the  foundation  prin- 
ciples of  the  two  great  divisions  of  elementary  mathe- 
matics, geometry  and  algebra  (but  in  particular  and 
chiefly  the  latter),  and  that  which  concerns  us  primarily 
in  the  work  of  Newton  is  his  introduction  of  the  con- 
tinuous variable  as  one  of  the  elements  of  algebraic 
analysis.  For  our  present  purpose  it  will  be  sufficient 
to  describe  the  continuous  variable  as  a  straight  line, 
having  one  of  its  extremities  fixed  at  a  point  O,  while 
the  point  P,  which  marks  the  other  extremity,  is  free  to 

O __P 

move  forwards  or  backwards  in  a  straight  line.  When 
P  moves  without  interruption  in  its  path  the  variable 
quantity  OP  is  said  to  change  continuously,  and  is 
called  a  continuous  variable.  It  may  or  may  not  be 
possible  to  represent  it  by  a  rational,  that  is,  a  com- 
mensurable number. 

At  last  the  materials  for  a  complete  grounding  of 
algebraic  science,  in  logically  fundamental  principles, 
were  at  hand.  Napier  had  given  us  logarithms,  Des- 
cartes had  put  into  our  hands  adequate  means  for  the 
graphical  representation  and  interpretation  of  the  alge- 
braic function,  Newton  had  shown  us  how  algebraic 
quantity  could  be  freed  from  the  limitation  by  which 
its  meaning  had  always  been  confined  to  rational 
number.  The  materials  for  the  work  were  indeed 
at  hand,  but  they  were  in  great  part  raw  materials, 
and  the  energies  of  mathematicians  were  expended 
upon  testing  and  reshaping  the  powerful  instruments 
of  analysis  just  discovered  and  enlarging  the  scope  of 
their  application,  and  little  attention  was  given  to  a 
reexamination  of  the  foundations  of  algebra.  Yet,  as  a 
science,   founded  upon  logically  scientific   principles, 


Elementary  Mathematics.  45 

algebra  was  still  in  the  formative  stage  and  required 
this  reexamination  in  the  light  of  the  new  discoveries. 
Two  centuries  have  been  just  sufficient  to  accomplish 
the  task. 

During  the  eighteenth  century  no  important  result 
was  attained.  The  old  algebra  had  acquired,  during 
the  centuries  since  the  revival  of  learning,  a  momen- 
tum sufficient  to  carry  its  development  forward  upon 
the  lines  that  had  been  originally  set.  But  at  the 
beginning  of  the  present  century  a  distinct  step  in 
advance  was  taken,  by  Argand  and  Gauss,  through  the 
introduction  of  the  so-called  imaginary  as  a  quantity, 
having  equal  importance  in  algebra  with  so-called  real 
quantity,  and  being  susceptible  of  real  geometrical 
interpretation.  This  step  made  it  possible,  as  soon  as 
'  its  importance  was  fully  understood,  to  define  algebra 
for  the  first  time. 

When  a  series  of  elements  operating  upon  each 
other  in  accordance  with  fixed  laws  produce  only  other 
elements  belonging  to  the  same  series,  they  are  said 
to  constitute  a  group.  Thus  all  positive  integers,  sub- 
jected only  to  the  processes  of  addition  and  multipli- 
cation, produce  only  positive  integers,  and  hence  form 
a  group. 

The  effect  of  introducing  into  the  arithmetic  of  posi- 
tive integers  the  further  processes  of  subtraction  and 
division  is  to  break  the  integrity  of  the  old  group  and 
form  a  new  one  whose  elements  include,  not  only  posi- 
tive integers,  but  all  rational  numbers,  both  positive 
and  negative,  integral  and  fractional.  A  final  step 
through  involution,  or  the  extracting  of  roots,  with  its 
allied  processes,  leads  in  a  similar  way  to  imaginary 
and  complex  numbers — that  is,  numbers  composed  of 
both  a  real  and  an  imaginary  part. 


46  Riverside  Addresses. 

Now  if,  as  is  legitimate,  we  regard  all  reals  and  imag- 
inaries  as  special  forms  of  complex  quantities — reals 
having  zero  imaginary  parts,  imaginaries  having  zero 
real  parts — then  the  algebraic  processes  of  addition, 
subtraction,  multiplication,  division,  evolution,  involu- 
tion, and  the  taking  of  logarithms,  applied  to  complex 
quantities  in  any  of  their  several  forms,  produce  only 
other  complex  quantities.     And  hence: 

The  aggregate  of  all  complex  quantities — includ- 
ing all  reals  and  imaginaries,  both  rational  and 
irrational — operating  upon  each  other  in  all  possible 
ways  by  the  rules  of  algebra,  form  a  closed  group. 

If,  in  an  algebra  the  elements  that  constitute  the 
subjects  of  its  operations  form  a  closed  group  when 
subjected  to  a  complete  cycle  of  such  operations,  such 
an  algebra  may  be  said  to  be  logically  complete. 

Now,  in  the  elementary  algebra  we  teach  in  the 
public  school,  involution  and  the  logarithmic  process 
form  an  essential  part,  and  through  them  imaginary 
and  complex  quantities  make  their  appearance  as  un- 
avoidable subjects  of  its  operations.  They  are  neces- 
sarily elements  coordinate  with  the  real  quantities  of 
our  algebra,  which  is  therefore  logically,  and  as  we 
now  agree,  also  practically,  only  the  fraction  of  an 
algebra,  if  the  complex  quantity  be  left  out  of  it.  The 
inability  of  the  earlier  algebraists  to  recognize  this 
fact  made  it  also  impossible  for  them  to  carry  out  the 
algebraic  processes  of  involution  and  the  taking  of 
logarithms  to  any  except  real  and  positive  numbers. 

Let  us  glance,  for  a  moment,  at  what  was  further 
necessary  in  order  to  complete  the  foundations  of  our 
mathematical  superstructure.  The  early  works  upon 
arithmetic  and  algebra  had  been  little  more  than  mere 
collections  of  rules  for  the  solution  of  problems.     Few 


Elementary  Mathematics.  47 

principles  were  explained,  none  adequately,  and  this 
feature  of  logical  incompleteness  has  remained  promi- 
nent, with  rare  exceptions,  in  all  text-books  upon 
algebra  up  to  the  present  time.  By  tradition,  algebra 
became  a  mere  mechanical  device  for  turning  out  prac- 
tical results,  by  careless  reasoning  errors  crept  into  the 
explanation  of  its  principles,  and  through  incompetent 
compilers  were  perpetuated  in  the  form  of  current  liter- 
ature, and  thus,  instead  of  becoming  a  classic  like  the 
geometry  handed  down  to  us  from  the  Greeks  in  the 
form  of  Kuclid's  elements,  algebra  became  a  collection 
of  processes  practically  exemplified  and  of  principles 
inadequately  explained. 

The  binomial  theorem  was  either  assumed  to  be  true 
for  negative  and  fractional  indices  by  mere  analogy 
from  one  or  two  special  cases,  or  a  logically  unsound 
proof  was  adduced  and  actually  remains  in  nearly  all 
of  our  text-books  to-day.  No  attempt  at  all  was  made 
to  justify  the  associative,  commutative,  and  distributive 
laws  in  the  four  fundamental  processes. 

Hence  the  necessity  of  a  thorough  overhauling  of  our 
algebraic  system  at  the  beginning  of  the  nineteenth 
century.  We  have  now  accomplished  the  task;  how 
well,  posterity  will  decide.  Abel,  in  1829,  examined 
thoroughly  and  laid  down  once  for  all  time  the  con- 
ditions under  which  development  by  the  binomial 
theorem  is  possible;  and  in  1870-71  Weierstrass  and 
G.  Kantor  gave  us  a  new  definition  of  irrational  num- 
ber, and  established  the  doctrine  of  the  irrational  upon 
a  strictly  logical  basis;  and  at  about  the  same  time 
(1870),  Benjamin  Peirce  produced  the  final  scientific 
formula  into  which  our  present  definition  of  algebra  is 
cast.  It  would  take  me  too  far  afield  to  go  over  these 
matters  here  in  detail.      They  require,  in  fact,  days, 


48  Riverside  Addresses. 

rather  than  hours,  of  careful  study  for  an  adequate 
understanding  of  them,  and  I  must  content  myself 
with  referring  you  to  the  literature  upon  this  part  of 
the  subject  under  discussion,  which  is  now  in  exist- 
ence and  easily  accessible. 

The  hurried  survey  we  have  now  made  of  the  course 
which  mathematical  thought  has  taken  within  the  two 
great  divisions  of  its  work  since  the  earliest  times, 
shows  how  at  every  step  in  the  progress  of  mathe- 
matical science,  whether  in  the  domain  of  algebra  or 
geometry?  it  has  seemed  impossible  for  the  mind  to  free 
itself  from  the  tendencies,  or  avoid  following  out  the 
line  of  development,  which  some  previous  age  or  gen- 
eration has  predetermined  for  it.  Slow  indeed  has  been 
our  progress,  if  we  merely  count  the  centuries  through 
which  the  struggle  against  difficulties  has  been  main- 
tained, though  the  achievement  has  undoubtedly  been 
very  great. 

But  standing,  as  we  do  now,  near  the  close  of  the 
twenty-fifth  century  since  mathematics  became  a 
science,  and  taking  advantage  of  the  discoveries  of 
our  predecessors,  we  may  pass  across  the  entire  field 
that  outlines  the  foundations  of  algebraic  science,  and 
seizing  only  upon  those  principles  that  mark  its  epochs 
of  advance  and  are  cardinal,  construct  an  algebraic 
system  which  shall  have  all  the  logical  rigor  and  com- 
pleteness of  Greek  geometry ;  and  for  the  accomplish- 
ment of  this  task  no  serious  difficulty  longer  stands  in 
our  way.  The  materials  are  at  hand  for  the  purpose; 
it  only  requires  some  master's  hand  to  mold  them  into 
coherence. 

Let  me  indicate,  in  the  briefest  possible  way,  what 
the  cardinal  principles  referred  to  are,  and  how  they 
constitute  the  foundations  of  algebra.  They  are  the 
following: 


Elementary  Mathematics.  49 

i.  The  doctrine  of  proportion,  first  formulated  by 
Pythagoras  (560  B.  C),  but  systematized  and  com- 
pleted by  Euclid  (300  B.  C).  This  includes  the  theory 
of  the  irrational,  and  leads  directly  to  the  definitions 
of  multiplication  and  division  in  algebra. 

2.  The  introduction  of  general  principles  or  rules  for 
the  fundamental  processes  of  arithmetic  and  algebra, 
by  the  Indians  in  the  early  centuries  of  the  Christian 
era. 

3.  The  introduction  of  letters  and  symbols  of  oper- 
ation, such  as  -f-j  — ,  X,  -5-,  =j  log.,  exponents,  etc. 
This  has  been  done  gradually,  by  many  hands. 

4.  The  invention  of  logarithms,  by  Napier  (16 14), 
out  of  which  the  law  of  indices  (exponents)  also  pro- 
ceeds. 

5.  The  graphic  representation  of  algebraic  functions 
(by  curves),  bjr  Descartes  (1637). 

6.  The  introduction  of  the  continuous  variable,  by 
Newton  (1665). 

7.  The  introduction  and  graphic  representation  of 
imaginary  quantities,  by  Argand  (1806). 

8.  The  final  definition  of  algebra,  due  to  the  general 
reconstruction  of  algebra  during  the  nineteenth  century, 
by  Grassmann,  Cay  ley,  Sylvester,  Peirce  (1 845-1 870). 

To  fill  in  this  outline  would  be  the  writing  of  a  book. 
Two  demonstrations  must  serve  the  purpose  of  giving 
you  a  hint  of  the  mode  of  procedure.  They  shall  be 
the  answers  to  the  two  questions: 

(a)  How  is  the  definition  of  multiplication  derived 
from  the  theory  of  proportion? 

(b)  How,  by  proper  definition  and  by  the  help  of  the 
graphic  method  and  the  continuous  variable,  can  we 
prove  the  law  of  indices  (exponents)? 

(a)  Multiplication. — The  propositions  of  proportion 
4— A 


50  Riverside  Addresses. 

are  assumed  to  be  true,  as  proved  by  Euclid,  not  by  the 
algebraic  method  used  in  American  text-books.  Let 
a  and  b  be  any  two  magnitudes  laid  off  upon  two 
straight  lines,  making  any  convenient  angle  with  each 
other.  At  O  lay  off  OB=£,  OA=«,  and  on  OA,  in  the 
same  direction  as  OA,  lay  off  OJ,  which  shall  be  of 
fixed  length  in  all  constructions  belonging  to  algebra, 
and  shall  be  called  the  unit.  Join  J  and  B,  and  draw 
from  A  a  straight  line  parallel  to  JB  to  intersect  OB 
in  M. 

Then,  by  the  theory 
of  proportion    in   the 
similar  triangles  OJB, 
OAM, 
j:b:  :a:m.   [;;/=OM]. 

The      line-segment 
OM=//z  is  defined  as  the  algebraic  product  of  a  by  b, 
and  is  denoted  by  aXb* 

The  law  of  commutation,  namely:  aXb—bXa,  fol- 
lows at  once  from  Euclid,  V.  16,  which  proves  that  if 
j'.b:  :a:vi 
:.    j:a:  :b:m, 
of  which  proportions  the  first  states  by  our  definition 

that 

m=aXb; 
the  second  that 

m—bXci' 

If  Di  be  equal  to  j,  then 

j:b:  :a:j 
and  b,  a  are  called  the  reciprocals  of  «,  b,  respectively; 
and  they  are  written  thus: 

j  I  <?=reciprocal  of  a, 
j  /  $=reciprocal  of  b,  etc. 

*  Descartes,  la  Geometrie  (1637),  reprint  of  1886,  p.  2. 


Elementary  MatJiematics.  51 

If  a=j  in  the  proportion/ :$:  :a:m,  then 
j'.b:  \j':m,  that  is  w=b; 
but  by  definition  m=jXb 

.-.    Jxb=b 
and  in  particular 

jXj=j, 
which  is  the  well-known  property  of  unity. 
Again,  the  proportion  j  :b  ::a\j  defines 

b=j  ja,   a=J/b,  aXb=J; 
hence, 

a  X  b=a  Xj  /  a=j  Xa  /  a=a  /  #=/, 

and  similarly 

bXa=b\b=j. 
Also, 

aXj/'b=jXa/b=--a/b. 

Thus  ^/^  is  a  fraction  or  quotient  in  the  algebraic 
sense. 

These  examples  suffice,  no  doubt,  to  convince  you 
that  all  the  laws  of  algebraic  multiplication  and  division 
can  be  deduced  as  direct  consequences  from  the  theory 
of  proportion  as  laid  down  by  Euclid. 

(b)  The  Index  Law. — Suppose  P,  Q  to  be  two  points 
moving  in  a  straight  line,  the  former  with  a  velocity 
proportional  to  its  distance  from  a  fixed  origin  O,  the 

o.J  P'  P" 

; — J — ' ; z 

q  tf  q" 

latter  with  a  constant  velocity.  L,et  x  represent  the 
variable  distance  of  P,  y  that  of  Q  from  the  origin,  and 
let  A  be  the  velocity  of  P  when  x=  i ,  f-i  the  constant 
velocity  of  Q.  The  velocities  that  P  will  have  acquired, 
as  it  arrives  at  the  positions  J,  P',  P",  respectively,  are 

A ,  x '  A ,  x  "  A , 
and  the  corresponding  values  of  x  andy  are 


52  Riverside  Addresses. 

.r=OJ=i,  ,r'=i+JP',  *"=i-KrP", 

Q  being  supposed  to  pass  the  origin  at  the  instant  when 
P  passes  J,  at  which  point  #==OJ=i.  The  velocity  of 
P,  relatively  to  that  of  Q,  is  known,  or  vice  versa,  as 
soon  as  the  ratio  of  ju  to  A  is  given.  By  means  of  this 
construction  the  terms  modulus,  base,  exponential,  and 
logarithm  are  defined  as  follows: 

(i)  /*/  A,  a  given  value  of  which,  say  m,  determines  a 
system  of  corresponding  distances  x',  x'\  ...  and/', 
y",  ...  is  called  the  modulus  of  the  system. 

(ii)  The  modulus  having  been  assigned,  the  value 
of  x,  corresponding  to  y=0]=i,  is  determined  as  a 
fixed  magnitude,  and  is  called  the  base  of  the  system. 
We  may  denote  it  by  b. 

(iii)  x,  x',  x"  .  .  .  are  called  the  exponentials  of  y, 
y',  y",  .  .  .  respectively,  with  reference  to  the  base  b, 
and  the  relation  between  x  and  y  is  written 
x=by. 

(iv)  y,  y,  y"  .  .  .  are  called  the  logarithms  of  x,  x', 
x",  ...  respectively,  with  respect  to  the  modulus  vi, 
and  the  symbolic  definition  is 

j,/=l0gm  x. 

Now,  by  virtue  of  definition  (iii), 

^'=i+JP',    ^"=i+JP" 

.-.  ^"/^'=op"/op' 

=i+P'P"/OP'. 
But  the  magnitude  i+P'P"/OP'  represents  the  dis- 
tance that  P  would  traverse,  starting  at  unit's  distance 
to  the  left  of  P',  on  the  supposition  that  its  velocity  at 
P'  is  A  x '  jx  '=  A,  and  the  corresponding  distance  passed 
over  by  Q  is  y" — y' ;  hence  by  definition  of  the  expo- 
nential, 

i+P'P"/OP'=£y"   r 


Elementary  Mathematics.  53 

and  therefore 

which  is  the  index  law  for  a  quotient.     In  particular, 

if/'=o, 

or  ilb?'=b-", 

and  therefore  writing  — -J/'=J/,  by"  I  bY'=by"— y'  becomes 

which  is  the  index  law  for  a  product. 

The  addition  theorem  for  logarithms,  namely: 

iog(m)  (/'x/)=iog(nv+  log^y, 

may  be  proved  in  a  similar  manner  from  the  figures,  or 
it  may  be  deduced  as  a  consequence  of  the  index  law, 
and  you  will  doubtless  be  willing  to  take  my  word  for 
it  that  the  law  of  involution 

(£h)k=(£k)h=£hk, 
together  with  its  inverse  logarithmic  process 

h  log  x*=k  log  x)x=hk  log  x, 
can  be  justified  by  the  methods  here  described  with 
equal  facility. 

You  will  doubtless  be  interested  to  know  that  Napier's 
original  definition  of  a  logarithm  was  in  its  essentials 
the  same  as  that  I  have  just  given. 

I  must  not  tax  your  patience  longer.  If  the  thoughts 
I  have  chosen  to  lay  before  you  this  morning  have  not 
touched  at  many  points  the  work  that  goes  on  in  your 
school-rooms  from  day  to  day,  I  console  myself  with 
the  hope  that  in  your  own  intellectual  life  outside  the 
mathematical  recitation-room  some  place  may  be  found 
for  the  cultivation  of  your  chosen  subject  as  a  science; 
for  a  thorough  examination  of  its  foundations,  for  a 
careful  study  of  its  history,  and'  of  the  life-work  of  the 
men  who  have  made  it  what  it  is. 


54  Riverside  Addresses. 

I  venture  to  predict  that  a  judicious  course  of  read- 
ing in  mathematics  would  have  many  beneficial  effects 
in  your  teaching.  An  anecdote  concerning  the  man 
who  first  propounded  the  pons  asmorum,  or  who  discov- 
ered the  binomial  theorem,  would  be  caught  up  by  your 
pupils  with  eager  interest.  The  great  discoveries  in 
science  are  comparable  with  the  great  crises  in  political 
history.  By  what  principle  in  education  do  we  teach 
our  children  to  revere  the  names  of  L,eonidas  and  War- 
ren, because  they  were  the  heroes  of  Thermopylae  and 
Bunker  Hill,  but  neglect  to  tell  them  to  whom  they 
are  indebted  for  the  knowledge  made  accessible  to  them 
in  their  arithmetic,  their  geometry,  and  their  algebra? 
I  recommend  as  subjects  worthy  of  study  and  discus- 
sion in  }rour  class-rooms,  sketches  in  the  history  of 
mathematics. 


Physics  in  Secondary  Schools.  55 


PHYSICS  IN  SECONDARY  SCHOOLS. 

SOME    ASPECTS    OF   THE    PRESENT    SITUATION. 
By  Frederick  Slate, 

Professor  of  Physics. 


In  appearing  before  this  gathering  to-day,  it  is 
pleasant  to  bring  with  me  the  thought  that  conditions 
as  regards  high  school  work  have  become  so  much 
more  favorable  during  the  five  years  that  have  slipped 
by  since  I  first  made  my  appearance  in  this  role;  and 
in  discussing  physics-teaching,  advocated  the  policy 
of  making  a  permanent  acquisition  of  our  conceded 
share  in  secondary  education  through  reform  of  method, 
and  clear  view  of  relation  to  other  parts  in  the  scheme 
of  training.  Such  ameliorations  as  the  present  status 
shows  are  due,  I  am  glad  to  be  able  to  say,  largely  to 
continuous  process,  and  not  altogether  to  the  stimulus 
which  the  acts  passed  by  the  last  legislature  applied. 

In  the  annual  tour  through  the  State,  which  has  come 
to  be  an  important  part  of  my  regular  duty,  the  gaps 
between  one  high  school  and  the  next  in  which  good 
work  in  all  branches  is  turned  out,  are  found  to  be 
closing  up  with  every  passing  year.  Vivid  interest  for 
the  best  ideas  is  keeping  the  workers  astir  in  centers 
now  numbering  ten  for  one  that  was  to  be  found  five 
years  ago;  and  physics  has  not  lagged  unduly  behind 
the  general  front  of  the  advance. 

The  University  of  California  made  an  early  begin- 
ning, as  compared  with  other  institutions  of  like  grade, 


56  Riverside  Addresses. 

in  laying  down  a  requirement  in  physics  for  matricula- 
tion. This  subject  was  introduced  as  alternative  with 
chemistry  or  botany  nearly  ten  years  ago.  In  1887 
free  option  among  the  group  ceased,  and  since  then 
physics  has  been  exacted  of  all  except  matriculants 
upon  the  classical  and  literary  requirements.  During 
these  later  years,  too,  the  examination-papers  set,  and 
what  other  influences  could  be  exerted,  were  all  con- 
sistently directed  towards  securing  thoroughness  of 
work,  correctness  of  method,  and  wise  selection  of  mate- 
rial. There  is  strong  satisfaction  in  recognizing  that 
the  ideas  of  this  kind  which  our  State  University 
championed  from  the  outset,  have,  at  this  date,  found 
general  acceptance.  "  Where  a  concession  is  made  to 
those  who  demand  some  science,  *  *  *  physics 
is  the  subject  selected,  because  it  includes  more  generic 
principles  than  any  other  science."  This  quoted  sen- 
tence carries  the  authority  of  the  committee  report  upon 
secondary  education  to  the  next  meeting  of  the  National 
Educational  Association  at  Saratoga,  in  July,  1892. 

While,  then,  there  is  cause  for  self-congratulation;  I 
mean  with  the  secondary  education  of  the  State,  not 
with  the  State  University  only;  it  is  nevertheless  not 
to  be  lost  sight  of  that  the  only  mental  attitude  con- 
sistent with  progress  is  that  of  cooperative  inquiry  and 
experiment,  in  order  to  sift  out  what  may  be  regarded 
as  established  among  the  ideas  and  processes  which 
have  been  tentatively  introduced  into  school- work;  to 
recognize  and  adopt  them,  and  thus,  while  we  assure 
our  foothold  upon  ascertained  truth,  to  narrow  down 
the  field  within  which  opinion  needs  to  waver  among 
uncertainties.  It  is  equally  inexpedient  to  look  upon 
our  problem  as  fully  and  permanently  solved,  or  to  think 
of  ourselves  as  still  merely  upon  the  threshold  of  it. 


Physics  in  Secondary  Schools.  57 

It  may  be  profitable,  therefore,  as  well  as  encouraging, 
if  we  here  put  down  in  brief  summary  some  items  in 
regard  to  which  our  position  to-day  is  distinctly  an 
advance  as  compared  with  that  which  we  were  com- 
pelled to  occupy  only  a  very  few  years  ago. 

First,  then,  we  need  no  longer  to  be  clamorous 
about  the  importance  of  natural  science  in  general  or 
physics  in  particular.  For  causes  apparent  to  most  of 
us,  some  of  which  will  be  alluded  to  in  another  con- 
nection presently,  several  features  of  this  work  as  done 
in  schools  have  made  such  strong  appeal  to  popular 
appreciation,  that  the  various  authorities  have  been 
easily  led  to  grant  freely  what  has  been  so  unanimously 
demanded.  There  was  a  time  when  I  was  obliged  to 
plead  for  physics ;  but  our  present  effort  can  be  turned 
to  making  wise  use  of  that  which  is  the  fruit  of  old- 
time  exertions. 

In  the  second  place,  the  broader  features  of  the 
methods  most  appropriate  to  the  study  have  been  stated 
and  generally  agreed  upon.  If  some  older  ones  of  us 
call  up  in  memory  the  days  when  we  studied  natural 
philosophy  as  we  learned  our  catechism — by  question 
and  answer  laid  down,  and  without  a  vestige  of  direct 
reference  to  the  realities  to  which  these  statements 
applied — such  memories  but  swell  the  tide  of  humorous 
pity  for  the  methods  in  which  the  earlier  generations 
were  schooled;  methods  so  perverse,  according  to  all 
psychology  and  wider  view  of  life,  that  we  are  reduced 
to  wonder  how  so  excellent  a  body  of  men  and  women 
could  have  been  shaped  in  so  bad  a  mold. 

And  again,  the  benefits  of  what  we  rightly  call  reform 
in  secondary  education  have  accrued  to  our  branch  of 
study  as  well  as  others.  We  have  cut  down  the  num- 
ber of  subjects  taken  up,  and  devoted  a  larger  fraction 


58  Riverside  Addresses. 

of  the  pupil's  time  to  each;  nor  is  there  reasonable 
doubt  that  we  shall  find  advantage  in  continuing 
this  process  beyond  any  point  yet  attained.  In  so 
doing  we  gain  opportunity  to  impart  what  I  call 
serious  knowledge;  to  develop  each  branch  to  the 
degree  necessary  for  making  permanent  impression 
upon  young  brains;  and  to  lift  ourselves  out  of  the 
slough  of  superficialness  into  which  high  schools  have 
been  brought  by  a  course  of  study  touching  upon 
almost  all  sides  of  learning. 

Indirectly,  too,  we  are  helped  by  the  demand  this 
thoroughness  in  what  is  taught  makes  upon  the  one 
who  teaches.  In  a  course  laid  down  according  to 
modern  ideas,  there  is  no  place  for  any  except  those 
well- trained  and  fully-prepared  to  teach  sound  knowl- 
edge on  the  lines  of  good  methods. 

With  place  assured,  plans  well  outlined,  and  an 
emphatic  demand  for  excellence  of  instruction,  we 
have  within  reach  good  leverage  for  further  progress; 
and  we  must  not  shrink  from  the  weighty  responsibil- 
ity of  keeping  our  "  intellectual  conscience"  awake  in 
its  search  for  still  better  things,  and  of  conforming  our 
practice  to  them  as  soon  as  found. 

What  I  have  further  to  say  is  to  be  thought  of  as 
my  endeavor  to  formulate  for  myself  and  for  you  the 
truth  about  what  are  now  either  (i)  unclearly-held 
views,  or  (2)  false  tendencies,  needing  to  be  combatted, 
or  (3)  elements  worthy  of  being  included  in  our  teach- 
ing. 

It  is  altogether  likely,  indeed,  that  some  of  you  have 
heard  from  me,  on  various  occasions  and  disconnectedly, 
all  that  the  allotted  time  allows  me  to  bring  forward 
here.  Nevertheless,  the  more  public  utterance  and 
discriminating  statement  seem  to  offer  advantages 
which  warrant  what  may  be  introduced  of  repetition. 


Physics  in  Secondary  Schools.  59 

For  some  years  past,  we  may  even  say  ever  since  its 
introduction  as  a  feature  of  school- work,  the  laboratory 
has  offered  themes  fruitful  of  thought,  and  fruitful  of 
discussion  because  of  divergence  in  thought.  It  will 
be  a  distinct  gain  if  we  may  contribute  aught  to-day 
towards  fixing  the  scope  and  limitations  of  laboratory 
work,  so  that  we  are  able  to  say  of  its  field  of  useful- 
ness: "  It  lies  thus  and  so,  and  is  hedged  by  these  plain 
boundaries."  If  we  can  in  any  fair  measure  accom- 
plish this,  there  will  follow  clearer  insight  into  the 
reason  for  its  existence  as  a  feature  of  science-teaching 
at  the  elementary  stage;  better  judgment  as  to  its 
necessity  or  indispensableness. 

It  would  not  surprise  me  greatly  to  discover  a  num- 
ber present  holding  a  discussion  of  these  points  to  be 
a  veritable  threshing  of  chaff;  but  the  very  diversity  of 
opinion  and  practice  proves  them  undecided  now,  nor 
would  it  be  anything  unique  if  we  should  find  that 
they  had  never  been  clearly  thought  out  by  many  of 
those  who  zealously  take  part  in  introducing  the  labo- 
ratory system.  For  it  is  rather  characteristic  of  the 
early  stages  in  reform  movements,  that  the  rank  and 
file  at  least  are,  in  the  literal  sense  of  the  word, 
adherents,  and  cleave  to  the  view  which  happens  their 
way,  while  the  privilege  of  holding  conviction  as  the 
result  of  thinking,  and  at  the  same  time  making  it  the 
basis  of  action,  is  reserved  to  the  few  who  are  akin  to 
the  generations  following  that  in  which  their  lot  is  cast. 

It  would  be  to  some  extent  true,  I  am  sure,  if  we 
should  trace  a  partly  imitative  process,  which  found 
the  University  laboratory  in  existence  with  its  function 
of  research,  modeled  the  college  laboratory  after  it,  and 
copied  this  feature  into  the  school  system,  without 
pausing  to  very  clearly  define  how  the  original  function 


60  Riverside  Addresses. 

must  be  modified  to  suit  the  changed  conditions.  This 
is  not  said  at  random;  I  have  been  met  on  many  occa- 
sions with  distinct  indications  of  this  sequence. 

Now  it  is  apparent  that  out  of  the  school-laboratory 
comes  no  investigation,  in  the  sense  of  "increase  of 
ascertained  truth."  Rediscovery  of  truth  by  the  pupil 
there  may  be;  ought  certainly  to  be,  and  to  a  large 
extent,  we  find  maintained,  and  we  shall  revert  to  this 
before  leaving  the  topic;  but  the  function  of  the  matured 
scientific  worker  does  not  yet  come  into  action. 

In  another  direction  there  are  signs  of  attempted 
alliance  with  entirely  different  tendencies.  About  three 
years  ago  the  idea  of  manual  training  received  a  strong 
impetus;  some  projects  were  even  entertained  and 
discussed  for  engrafting  it  upon  the  common  school  sys- 
tem. Did  we  not  note  movements  of  attraction  between 
laboratory  and  manual  training  shop?  It  would  be 
wrong  to  den}'  the  existence  of  common  ground  on 
which  these  two  different  ideas  overlap.  But  coales- 
cence would  not  have  been  in  the  true  interest  of  either, 
and  we  must  all  have  been  glad  to  see  the  danger  fade 
away. 

I  spoke  freely  on  this  subject  at  the  Sacramento 
meeting  in  1888;  if  I  make  allusion  to  it  here,  it  is  for 
the  purpose  of  illustrating  how  a  thing  may  be,  and 
yet  those  who  are  called  upon  to  administer  it  be 
wrapped  in  mistiness  as  to  its  true  mission.  The  well- 
balanced  mind  avoids  confounding  these  matters,  as  the 
eye  escapes  the  confusions  of  color-blindness;  by  having 
within  it  faculty  for  perceiving  all  primary  elements, 
and  distinguishing  shades  and  tints  as  blendings  of 
these  in  varied  proportion. 

Well,  if  the  essential  of  laboratory  work  in  schools 
be  not  manual  training,  that  is,  apparatus  building,  and 


Physics  in  Secondary  Schools.  61 

cannot  be  discovery  of  new  truth,  may  it  not  perhaps 
be  new  discovery  of  truth?  Shall  the  classes  be  so 
conducted  that  the  pupils  are — to  put  the  extreme 
case — told  to  make  an  experiment,  being  left  unin- 
formed how  the  result  should  fall  out  ?  I  could  quote 
very  respectable  authority  for  the  criticism  upon  cer- 
tain books:  "Their  scheme  is  all  wrong,  because 
they  tell  the  pupil  beforehand  what  to  expect."  That 
the  proposed  plan  quickens  observation,  cultivates  the 
investigating  habit  of  mind,  and  lends  keen  zest  to  old 
experiments,  although  the  sweetness  of  discovery  is 
but  tasted  with  the  tip  of  the  tongue,  and  is,  after  all, 
only  of  the  kind  we  see  in  our  boys  as  they  hang 
breathless  over  Ivanhoe  or  Robinson  Crusoe — all  this 
must  be  readily  conceded  b3'  every  candid  mind.  There 
is  much  that  appeals  to  me  in  an  inclusive  view  of 
education  as  a  thinking  again  of  the  thoughts  of  the 
race;  with  many  foreshortenings,  to  be  sure;  with  ex- 
tensive elimination  of  periods  of  groping,  and  much 
redistribution  of  emphasis;  but  in  the  main  a  bringing 
into  contact  with  the  recorded  thought  of  humanity. 
So,  too,  there  is  of  necessity,  under  circumstances  as 
sketched  above,  much  of  that  individual  treatment  for 
pupils  which  is  a  marked  characteristic  of  best  educa- 
tion; to  be  striven  after  at  every  point  among  the 
elements  opposed  to  it  which  the  handling  of  crowded 
classes  entails. 

If  we  tone  down  the  leading  idea  from  rank  extreme 
to  somewhat  of  moderation,  I  can  fancy  such  a  charm- 
ing picture  of  wise  teacher  in  the  midst  of  alert  pupils, 
each  actively  thinking  for  himself,  and  aided  to  the 
extent  and  in  the  way  suggested  by  the  obstacles  that 
block  his  path,  that  I  am  willing  to  let  this  ideal  of 
laboratory  stand  bright  before  us,  undimmed  by  a.\\y 


62  Riverside  Addresses. 

critical  analysis.  Pupils  may  be  led  from  stepping- 
stone  to  stepping-stone  with  proper,  half-dissimulated 
guidance,  avoiding  treacherous  footholds,  and  thus 
springing  nimbly  and  securely  over  many  a  gap  where 
the  path-finder  stumbled  and  fell  short. 

Let  our  further  consideration  and  argument,  there- 
fore, be  confined  to  examining  what  results  may  be 
gained  by  executing  a  plan  like  this,  and  seeing  in 
how  far,  as  well  as  in  what  directions,  such  procedure 
needs  supplementing  before  the  requirements  of  the 
case  are  fully  met.  An  examination  of  this  sort  cannot 
but  prove  fruitful  if  its  brings  out  the  truth,  uncon- 
sciously possessed  it  may  be,  which  gives  to  our  plans 
and  ideas  their  real  value — value  often  ascribed  to 
widely  different  sources. 

The  word  "  rediscovery  "  seems  to  me  apt  as  suggest- 
ing a  feature  in  the  working  of  the  pupil's  mind  akin 
to  what  we  find  historically  true  in  the  process  of 
original  discovery.  I  mean  that  the  results  obtained 
involve  a  large  element  of  intuition,  and  are  not  con- 
clusive and  unassailable,  as  they  may  become  through 
later  scrutiny  and  test.  It  is  strikingly  shown,  as  we 
follow  the  course  of  development  in  the  work  of  men 
like  Galileo,  like  Huyghens,  like  Mayer,  even,  how 
incompletely  established  certain  most  valuable  results 
were  at  the  date  of  their  first  announcement.  As  in 
Alpine  climbing  the  leader  on  the  rope  is  often  obliged 
to  content  himself  with  precarious  foothold  and  slippery 
grasp  as  he  picks  the  way,  while  those  behind  him 
secure  themselves  in  notches  cut,  so  Newton's  minute 
examination  for  accuracy  is  needed  to  confirm  Galileo's 
results;  Young's  labors  bring  testimony  in  support  of 
Huyghens's  generalizations,  and  only  through  Joule's 
patient  detail  comparisons  is  full  acceptance  gained  for 
Mayer's  ideas. 


Physics  in  Secondary  Schools.  63 

At  the  very  best  our  pupils  repeat  upon  a  minute 
scale  the  experiences  of  the  former  group,  not  those  of 
the  names  in  the  second. 

We  hear  much  said  of  the  "  inductive  method,"  which 
is  often  understood  in  a  way  that  may  be  characterized 
as  pretending — with  more  or  less  truth  admixed — that 
we  do  not  know;  but,  as  a  matter  of  fact,  hypothesis 
generally  precedes  experiment,  and  we  test  in  order 
to  discover  whether  we  do  know.  Jevons  has  clearly, 
and  as  I  think  justly,  pointed  out  how  essential  to  the 
real  process  is  this  rapid  flashing  back  and  forth  between 
assumption  and  putting  it  to  the  proof. 

Have  we  not  all  in  mind  Faraday's  form  of  stating 
the  truth?  "  You  must  tell  me  what  I  am  expected  to 
see,"  he  said,  "before  I  can  know  whether  I  see  it 
or  not."  That  is,  the  mind  must  be  prepared  by 
Irypothesis;  we  must  look  at  phenomena,  not  with 
dispersed  bovine  gaze,  but  with  retinal-spot  highly 
sensitized  for  the  one  thing  among  the  mass  which  are 
in  sight.  Ability  to  prophesy  is  the  criterion  of  true 
knowledge.  The  bearing  of  these  remarks  will  be  plain, 
I  hope,  when  we  remember  that  our  young  people  are 
doing  their  work  under  similar  conditions,  and  in  addi- 
tion are  limited  by  their  feebler  powers. 

We  must  not  expect,  then,  that  laboratory  practice, 
even  for  a  favored  group,  is  going  to  emancipate  from 
all  reliance  upon  authority,  and  enable  us  to  write  as 
device:  "Accept  only  what  you  have  yourself  worked 
out;"  or  even  the  modified  form,  "What  you  have 
yourself  thought  out."  By  occupying  such  extreme 
positions,  which  are  in  reality  untenable,  we  constrain 
ourselves  into  adopting  for  defense  controversial  meth- 
ods, and  losing  that  candor  of  mind  which  lies  open  to 
the  truth. 


64  Riverside  Addresses. 

In  the  light  of  many  years  spent  in  teaching  and 
thinking,  it  seems  impossible  to  affirm  that,  try  as  we 
may,  there  can  be  eliminated  from  our  teaching  the 
element  of  acceptance  upon  authority.  Beginners  must 
continually  take  bearings  as  they  advance,  and  find 
whether  they  are  in  the  beaten  path,  as  I  am  told  the 
geological  survey  runs  its  traverse-lines  with  repeated 
adjustment  to  the  previously  established  points  of  an 
accurate  triangulation.  And  ought  this  element  to  be 
cast  out  root  and  branch  as  wholly  bad?  Is  not  the 
claim  that  it  should  based  upon  extreme  reaction  against 
preponderant  dogmatism?  It  is  true,  rather,  that  a 
part  of  all  sound  education  is  to  cultivate  due  reverence 
for  the  past,  and  esteem  for  the  legacies  handed  on  to 
us  from  its  worthies.  The  great  and  the  good  have 
worked  to  little  purpose,  if  we  are  to  refuse  their  inherit- 
ance unless  every  coin  of  it  has  been  tried  upon  each 
individual  touchstone.  Nor  should  we  fail  to  see  the 
artificialness  of  a  system  which  sets  up,  within  a  small 
range  and  during  a  limited  period,  standards  whose 
use  cannot  be  extended  in  time  or  space  to  cover  human 
life.  It  is  true  of  mature  men  of  widest  power  that  they 
command  with  independent  research  a  small  arc  only 
upon  the  circumference  of  knowledge. 

In  entering  this  plea  for  a  remainder  of  authoritative 
teaching,  I  am  running  a-tilt  against  no  mere  man  of 
straw.  It  is  unfortunately  true  that  just  such  extrav- 
agant claims  as  are  here  implied  are  made  for  the  kind 
of  work  that  we  are  discussing,  which  tend  to  bring  it 
into  disrepute  with  heads  clear  enough  to  see  the  fallacy 
of  the  position  assumed.  Interest  in  the  adoption  of 
the  best  methods  demands  that  we  should  avoid  such 
extremes  as  have  been  held  up  for  criticism.  We  accept 
the  law  of  gravitation,  or  the  law  of  reflection  for  light, 


Physics  in  Secondary  Schools.  65 

or  Ohm's  law  for  electrical  currents;  in  fact  the  great 
bod}'  of  laws  which  constitute  what  may  be  called 
formal  physics — laboratory-work  of  school  grade  can 
at  most  think  consequences  of  these  and  try  to  real- 
ize them.  As  a  recent  writer  has  said  (and  I  think 
his  utterances  are  the  best  I  have  seen  in  print  upon 
the  subject,  perhaps  because  thej^  come  in  as  inde- 
pendent corroboration  of  what  I  have  long  held  and 
preached):  *"The  laws  and  principles  which  have 
been  most  carefully  studied  by  scientific  men  should 
be  made  the  instruments,  not  the  objects,  of  scientific 
research.  The  teacher  should  avoid,  as  far  as  possible, 
experiments  whose  ostensible  object  is  to  establish  well- 
known  facts,  like  the  law  of  conservation  of  energy, 
the  truth  of  which  is  not  really  in  question.  But  the 
use  of  the  experimental  method,  as  an  illustration  of 
such  laws,  is  not  to  be  denied.  It  is  only  through  the 
aid  of  definite  examples  that  most  persons  can  arrive  at 
an  understanding  of  physics." 

A  trenchant  blow  is  struck  in  these  sentences,  every 
phrase  of  which  counts,  against  the  central  fallacy  of 
much  that  is  carried  out  in  school-work  as  quantitative 
measurement.  It  cannot  be  regarded  as  proving  the 
law;  but  it  is  a  proving  of  the  pupil;  of  his  conscien- 
tiousness, definiteness  of  knowledge,  and  power  to  take 
pains.  Here  lies,  indeed,  the  connecting  link  between 
manual  training  and  laboratory  work,  especially  of 
the  quantitative  kind.  Curiously  enough  this  bond 
attaches  on  what  we  may  denominate  the  ethical  side. 
The  constructive  use  of  principles  or  materials  has  a 
reactive  high  moral  value  when  wrong  figures,  or  errors 
cut  out  in  wood  or  iron,  bring  us  face  to  face  with 
faults  in  knowing  or  executing.     A  shrewd  friend  said 

*  Whiting,  Physical  Measurement,  p.  599. 
5— A 


66  Riverside  Addresses. 

to  me  many  years  ago,  that,  during  the  period  when 
we  are  fostering  the  thinking  faculty  in  the  younger 
generation,  those  subjects  are  the  best  adapted  tools, 
which  give  the  most  frequent  and  accessible  check  and 
control  upon  the  results  of  their  essays  at  thinking. 
This  is  the  kernel  of  the  truth,  I  am  sure.  It  may  be 
useful  to  the  unification  of  our  thought  to  see  a  large 
share  of  the  educational  value  in  such  subjects  as 
geometry  l}*ing  within  these  lines.  There  we  have 
flawless  standards,  it  is  true,  in  hypothetical  assump- 
tions, whose  discord  or  harmony  with  conclusion  gives 
certain  answer  as  to  correctness  of  reasoning;  but  laws 
of  physics  involved  in  school  work  are  scarcely  behind 
geometrical  axioms  in  practical  freedom  from  doubt. 
I  am  mistaken  if  some  of  the  readiness  with  which 
physics  has  been  incorporated  into  our  schemes  of 
study  is  not  due  to  the  recognition  that  it  furnishes  a 
large  and  new  available  supply  of  such  material. 

It  seems  true,  in  spite  of  the  claim  that  science-work 
in  schools  is  inductive  (meaning  exclusively  so),  that 
we  must  make  allowance  for  the  existence  of  a  con- 
siderable element  which  offers  the  essential  feature  of 
deduction.  Namely,  that  certain  ideas  are  assumed; 
with  a  coloring  of  reason  offered,  to  be  sure,  but  nec- 
essarily without  critical  analysis,  weighing,  and  discus- 
sion; and  of  these  the  pupil  thinks  a  corollary  and 
works  it  out  experimentally.  As  far  as  this  kind  of 
mental  exercise  is  concerned,  there  is  little  to  choose 
between  laboratory  problem  and  one  set  in  geometry. 

This  connection  leaves  the  door  open  to  place  a  word 
in  defense  of  qualitative  experiments,  which  it  is  the 
present  tendency  to  reject  almost  entirely.  But  the 
intuitive  activity  of  the  mind  finds  exercise  chiefly  in 
qualitative  work.     It  is  here  that  rediscovery,  with  its 


Physics  in  Secondary  Schools.  67 

(rather  fictitious)  novelty,  can  be  properly  spoken  of. 
I  know  that  students  in  my  own  classes  often  show  the 
onesidedness  that  is  the  fault  of  familiarity  with  those 
experiments  exclusively  which  are  employed  because 
they  afford  exact  methods,  while  experiments  are  passed 
by  that  yield  only  views  of  connection  between  cause 
and  effect  in  phenomena.  In  the  school-years  such 
exclusiveness  is  still  less  in  place,  so  that  this  thought 
deserves  of  us  that  we  bear  it  in  mind  and  correct  our 
practice  by  it. 

On  gathering  together  what  these  considerations  have 
thus  far  shown,  we  shall  have  before  us  the  answer,  as 
I  conceive  it,  to  the  question  which  concerns  itself  with 
results  to  be  expected  of  our  ideal  laboratory.  Those 
who  are  subjected  to  its  discipline  may  be  guided  to 
infer  the  qualitative  statement  of  physical  laws;  they 
may  experimentally  learn  to  know  what  are  the  main 
determining  causes  of  a  wide  range  of  phenomena. 
Their  conclusions  will  be  compared  with  established 
truth  mainly,  and  their  correct  understanding  of  rela- 
tions will  be  brought  out  by  measurements  which 
involve  the  application  of  assumed  quantitative  laws. 
As  bye-products  they  will  have  gained  a  fair  degree  of 
manipulative  skill — manual  dexterity,  that  is;  and,  if  it 
be  fair  to  call  this  a  bye-effect,  a  set  towards  conscien- 
tious accuracy  and  the  facing  of  facts  that  ramifies 
widely  into  the  fibers  of  character.  And  all  of  this 
with  healthful  stimulus  of  interest  and  development  of 
faculty  upon  the  two  great  lines  of  inductive  and 
deductive  reasoning. 

We  ma>'  be  content  to  call  this  a  goodly  harvest, 
although  our  claims  are  modest  by  the  side  of  those  I 
have  heard  advanced.  Training  like  this  is  desirable, 
and  that  although  the  actual  profit-sheet  may  never 


68  Riverside  Addresses. 

show  as  large  dividends  as  these.  Our  minds  are 
prepared  to  meet  a  coming-short  of  ideals  in  any  edu- 
cational scheme  as  applied  to  a  particular  group  of 
pupils;  but  it  is  of  the  highest  importance  to  know 
whither  our  effort  should  tend,  and  to  hold  ourselves 
and  our  young  charges  up  to  aim  for  the  very  best. 

Another  branch  of  our  inquiry  raises  the  issue 
whether  such  laboratory  training  is  likely  to  be  suffi- 
cient, or  whether  it  needs  supplementing.  Experience 
seems  to  show  that  it  does.  I  will  try  to  present  the 
matter  through  an  analogy.  The  suggestion  that 
comes  from  several  parallel  cases  is  of  the  same  tenor, 
and  we  should  heed  it.  Time  was,  when  the  study  of 
the  classics  was  prepared  for  by  solid  digging  at  the 
grammar,  which  was  mastered  by  rote  before  any 
venture  was  made  in  reading  the  literature.  This 
period  was  succeeded  by  a  strong  reactionary  tendency 
to  abolish  the  study  of  formal  grammar,  and  to  derive 
all  such  material  from  the  text.  It  has  been  found, 
however,  that  the  same  high  level  of  attainment  cannot 
be  reached  if  the  grammar-book  be  dispensed  with.  It 
is  introduced  at  a  different  stage,  in  less  isolated  fashion, 
with  larger  proportion  of  text-reading,  and  truer  expo- 
sition of  the  principles  of  grammar,  as  inferred  from  the 
observed  phenomena  of  the  language  used;  but  no 
substitute  completely  replaces  it. 

I  hope  for  the  day  when  the  school-grammar  of 
physics  shall  be  wisely  written,  fully  mastered  in  con- 
nection with  our  reading  of  the  text,  which  is  Nature, 
with  proper  understanding  of  the  statement  of  formal 
law  as  general  truth  obtained  by  collation  of  phenomena. 
We  have  had  our  days  of  rote-learned  grammar;  we 
have  had  our  phase  of  undue  reliance  upon  text-reading 
and  consequent  slovenliness.  Perhaps  better  things 
lie  just  ahead. 


Physics  in  Secondary  Schools.  69 

There  is  at  the  present  time  a  well-marked  drift 
towards  removing  difficulty  from  the  pathway  to 
knowledge,  which  gives  a  kind  of  general  support  to 
the  belief  which  has  spread  abroad  in  regard  to  labora- 
tory methods,  that  they  make  subjects  entertaining  to 
the  degree  of  being  learned  as  play.  There  should  be 
rejoicing  over  the  disappearance  of  those  difficulties 
which  have  their  source  in  allowing  the  end  to  be 
eclipsed  behind  the  means  to  the  end,  in  perverse 
routine,  and  the  whole  apparatus  of  pedantry;  that  evil 
spirit  whom  we  should  all  be  glad  to  know  of  as  exor- 
cised from  our  school-rooms  forever.  But  clearing  away 
this  rubbish  does  no  more  than  render  it  possible  to 
concentrate  effort  upon  the  real  inherent  resistance  of 
the  subject,  as  the  felling  of  a  tree  is  facilitated  by 
removing  the  brush  which  deadens  the  axe-blows. 
Difficulty  must  continue  to  meet  those  who  attack 
whatever  possesses  sterling  educational  value;  it  is,  in 
fact,  one  stamp  in  the  hall-mark. 

I  do  most  sincerely  maintain  that  we  should  put 
work  before  our  young  people  which  requires  effort, 
strenuous  effort,  in  order  to  accomplish  the  tasks,  set. 
We  must  continue  to  place  a  high  value  upon  the 
years  of  adolescence;  the  period  of  rapid  expansive 
growth  and  strong  assimilating  power  for  mind  as  well 
as  body;  the  time  during  which  the  habit  of  consecu- 
tive effort  should  become  ingrained.  We  can  do  no 
better  service  to  pupils  nearing  the  end  of  their  high 
school  course,  than  to  claim  and  gain  for  them  leisure 
to  satisfy  the  requirement  of  thinking,  and  growing 
strong  in  controlled  power  to  think,  while  we  discour- 
age nervous  cram  and  sham  acquisition. 

If,  then,  to  turn  from  general  statement  to  the  matter 
in  hand,  there  are,  as  I  have  tried  to  show,  needs  for 


70  Riverside  Addresses. 

systematic  knowledge  of  principles,  rooted  in  the  sub- 
ject-matter; existing,  not  as  alternative  to  laboratory- 
work,  but  as  condition  that  an  important  part  thereof 
may  be  successfully  prosecuted;  it  is  acting  in  the 
interest  of  the  pupil  if  we  do  not  shirk  the  task  thus 
imposed,  of  encountering  a  citadel  of  difficulty  transcend- 
ing those  met  in  the  recreative  experimental  outworks. 
We  are  thus  helping  to  inculcate  the  lesson  which  is 
the  quintessence  of  so  many  lessons;  that  is,  in  Hux- 
ley's words :  "  The  most  valuable  result  of  all  education 
is  the  ability  to  make  yourself  do  the  thing  you  have 
to  do,  when  it  ought  to  be  done,  whether  you  like  it  or 
not.  *  *  *  However  early  a  man's  training  begins,  it 
is  probably  the  last  lesson  that  he  learns  thoroughly." 
Our  newest  text-books  do  not  help  us  as  they  might 
just  here.  Their  abstruseness  has  been  largely  toned 
down  by  the  wholesome  and  clarifying  influences  of 
experimental  exercises.  So  much  thought  is  excited 
by  what  is  taken  in  hand,  and  so  much  explanation 
called  for  of  every-day  occurrences,  that  the  elimination 
of  advanced  theories  may  be  relied  upon  to  work  almost 
automatically.  We  can,  if  necessary,  aid  this  process  by 
judicious  omissions;  but  the  lax  incoherency  of  some 
of  our  books  is  not  so  easily  neutralized. 

We  are  all  aware  that  text-books  are  by  no  means 
mirrors  reflecting  opinion  current  among  experienced 
teachers;  so  it  would  be  unfair  to  judge  practice  by 
them.  All  teachers,  and  the  best  teachers  most  of  all, 
•  make  discriminating  use  of  their  books,  to  the  extent 
often  of  recasting  the  matter  in  presentation.  But 
print  is  powerful,  and  not  yet  altogether  divested  of 
sacredness;  the  molding  influence  of  the  book,  too,  is 
felt  most  strongly  by  those  very  younger  members  of 
the   profession   upon   whose  development   it  is  most 


Physics  in  Secondary  Schools.  71 

urgent  to  impress  the  right  direction.  It  seems  per- 
missible, therefore,  that  I  should  devote  to  criticism  of 
our  text-books  a  few  further  words,  which  are  a  protest 
against  the  lack  of  proportion  in  their  treatment  of 
topics;  and  in  saying  this  I  have  electricity  quite  promi- 
nently before  my  mind.  It  may  look  paradoxical  to 
raise  outcry  against  magnifying  those  applications  of 
physics  which  have  brought  an  influx  of  students — 20 
for  1  fifteen  years  ago — and  have  unlocked  coffers  for 
the  building  of  laboratories,  besides  forming  the  "  crown- 
ing success  of  the  century,"  as  they  have  harnessed  a 
new  form  of  energy  in  the  service  of  man.  Such  plea 
comes  with  all  the  more  force  and  temptation,  because 
the  time  in  which  we  are  sits  dazzled  under  the  sway 
of  electricity.  We  no  longer  realize  perspective  in 
these  things ;  but  physics  is  in  danger  of  being  swamped 
by  the  unexampled  material  success  in  this  one  of  its 
divisions,  as  an  examination  of  our  colleges  easily 
shows.  And  so  in  the  schools,  possession  is  taken  of 
our  books  from  the  early  pages,  in  which  centimeter- 
gram-second  system  is  forced  upon  pupils  unable  to 
comprehend  it,  to  the  Volt,  Ampere,  Wheatstone  bridge, 
and  other  members  of  the  goodly  company  that  bring 
up  the  rear. 

Application  of  principle  to  cases  occurring  in  daily 
life  meets  with  my  strongest  approval.  I  can  say,  in 
the  simple  words  of  Maxwell: 

*  "  Science  appears  to  us  with  a  very  different  aspect 
after  we  have  found  out  that  it  is  not  in  the  lecture- 
room  only,  and  by  means  of  the  electric  light  projected 
on  a  screen,  that  we  may  witness  physical  phenomena, 
but  that  we  may  find  illustrations  of  the  highest  doc- 
trines of  science  in  games  and  gymnastics,  in  traveling 

*  Scientific  papers,  Vol.  II,  p.  243. 


72  Riverside  Addresses. 

by  land  and  by  water,  in  storms  of  the  air  and  of  the 
sea,  and  wherever  there  is  matter  in  motion. 

"This  habit  of  recognizing  principles  amid  the  end- 
less variety  of  their  action  can  never  degrade  our  sense 
of  the  sublimity  of  nature,  nor  mar  our  enjo}-ment  of 
its  beauty.  On  the  contrary,  it  tends  to  rescue  our 
scientific  ideas  from  that  vague  condition  in  which  we 
too  often  leave  them,  buried  among  the  other  products 
of  a  lazy  credulity,  and  raise  them  into  their  proper 
position  among  the  doctrines  in  which  our  faith  is  so 
assured  that  we  are  ready  at  all  times  to  act  upon 
them." 

But  when  one  most  modern  application  is  seen  over- 
shadowing all  others  with  a  predominance  due  to 
evanescent  causes,  I  feel  we  are  making  a  blunder  if 
we  yield  to  the  pressure.  For  physics  is  the  classic 
among  the  sciences;  it  forms  the  permanent  record  of 
man's  success  in  accumulating,  through  centuries  of 
effort,  a  reasoned  explanation  of  large  groups  of  phe- 
nomena. Instead  of  reserving  chief  emphasis  for  the 
most  modern  contributions,  and  thus  ignoring  an  oppor- 
tunity, which  is  so  plainly  ours,  of  casting  off  the 
reproach  urged  against  science;  that  it  is  lacking  in  the 
human  element;  we  should  aim  at  bringing  out  the 
historical  aspect  of  discoveries  as  they  come  up  for 
treatment,  beyond  the  mere  uncommented  mention  of 
a  name,  and  at  exciting  lively  interest  in  this  picture 
of  international  cooperation  upon  an  edifice  which  has 
been  building  through  more  centuries  than  Cologne 
cathedral,  and  which  is  more  imposing  to  the  mental 
vision,  although  less  substantial  than  stone  and  mortar. 
I  recommend  to  you  sketches  from  the  history  of 
physics  as  valuable  material  for  the  class-room — well 
adapted  to  enforce  the  lessons  most  desirable  to  impress; 


Physics  in   Secondary  Schools.  73 

of  painstaking  slow  advance,  and  bard  work  over  every 
inch  of  conquered  ground. 

At  several  points  in  what  precedes,  I  have  implied 
the  close  relation  among  the  several  branches  of  high 
school  work  by  insisting  upon  the  results  of  comparing 
them  in  parallel.  Before  bringing  these  remarks  to  a 
close,  I  should  like  to  urge  this  idea  more  explicitly,  as 
its  value  deserves. 

The  task  of  the  public  school  on  the  intellectual  side 
may  be  considered  as  being  to:  (i)  Impart  useful 
knowledge;  (2)  Promote  clearness  of  thought;  (3) 
Cultivate  accurate  expression.  We  should  find  earnest 
striving  after  these  three  things  in  every  class-room. 
Where  this  is  the  case  with  studies  in  language,  math- 
ematics, and  science,  each  pair  buttresses  the  third,  and 
the  unity  of  purpose  sets  its  stamp  upon  the  pupil's 
mind. 

For  reasons  connected  mainly  with  the  need  of 
imparting  varied  and  extended  information,  we  segre- 
gate into  separate  class-rooms,  and  place  in  charge  of 
teachers  specially  qualified  to  handle  and  expound  por- 
tions of  the  material  which  it  is  advisable  to  treat.  Yet, 
after  all,  in  our  thought  and  in  their  essence,  these  are 
but  fractions  of  a  unity ;  these  comings  and  goings  and 
modulations  of  key  can  be  grouped  into  a  harmony  of 
movement  and  of  utterance  for  one  central  theme. 

Last  Spring,  an  eminent  preacher  and  lecturer  devel- 
oped for  me,  in  conversation,  the  idea  that  all  our  out- 
door sports  are  wrong  in  principle,  because  they  involve 
a  striving  of  one,  or  of  one  crew,  for  victory,  which 
must  also  mean  defeat.  The  truer  human  end  would 
be  reached,  he  maintained,  by  including  in  one  effect 
the  efforts  of  all,  combined  with  proportion  and  rhythm. 
There  is  perhaps  too  much  of  the  Anglo-Saxon  in  me 
6— A 


74  Riverside  Addresses. 

to  quite  accept  this  view  as  to  field-sports.  I  am 
not  convinced  that  rivalry  in  contest  is  not  succeeded 
by  generosity  in  those  who  excel,  ungrudging  praise 
in  those  who  are  proved  inferior.  But  within  the 
range  of  education  I  can  find  deep  truth  in  this  view. 
and  do  regard  it  as  bringing  to  utterance  soundest 
doctrine. 


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